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Theorem List for Metamath Proof Explorer - 34501-34600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhba1-o 34501 The setvar 𝑥 is not free in 𝑥𝜑. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (New usage is discouraged.)
(∀𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremaxc4i-o 34502 Inference version of ax-c4 34488. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)
(∀𝑥𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)

Theoremequid1 34503 Proof of equid 1985 from our older axioms. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and requires no dummy variables. A simpler proof, similar to Tarski's, is possible if we make use of ax-5 1879; see the proof of equid 1985. See equid1ALT 34529 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑥

Theoremequcomi1 34504 Proof of equcomi 1990 from equid1 34503, avoiding use of ax-5 1879 (the only use of ax-5 1879 is via ax7 1989, so using ax-7 1981 instead would remove dependency on ax-5 1879). (Contributed by BJ, 8-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremaecom-o 34505 Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 2344 using ax-c11 34491. Unlike axc11nfromc11 34530, this version does not require ax-5 1879 (see comment of equcomi1 34504). (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaecoms-o 34506 A commutation rule for identical variable specifiers. Version of aecoms 2345 using ax-c11 34491. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)

Theoremhbae-o 34507 All variables are effectively bound in an identical variable specifier. Version of hbae 2348 using ax-c11 34491. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Theoremdral1-o 34508 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 2356 using ax-c11 34491. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremax12fromc15 34509 Rederivation of axiom ax-12 2087 from ax-c15 34493, ax-c11 34491 (used through dral1-o 34508), and other older axioms. See theorem axc15 2339 for the derivation of ax-c15 34493 from ax-12 2087.

An open problem is whether we can prove this using ax-c11n 34492 instead of ax-c11 34491.

This proof uses newer axioms ax-4 1777 and ax-6 1945, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 34488 and ax-c10 34490. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax13fromc9 34510 Derive ax-13 2282 from ax-c9 34494 and other older axioms.

This proof uses newer axioms ax-4 1777 and ax-6 1945, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 34488 and ax-c10 34490. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

20.23.3  Legacy theorems using obsolete axioms

These theorems were mostly intended to study properties of the older axiom schemes and are not useful outside of this section. They should not be used outside of this section. They may be deleted when they are deemed to no longer be of interest.

Theoremax5ALT 34511* Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

(This theorem simply repeats ax-5 1879 so that we can include the following note, which applies only to the obsolete axiomatization.)

This axiom is logically redundant in the (logically complete) predicate calculus axiom system consisting of ax-gen 1762, ax-c4 34488, ax-c5 34487, ax-11 2074, ax-c7 34489, ax-7 1981, ax-c9 34494, ax-c10 34490, ax-c11 34491, ax-8 2032, ax-9 2039, ax-c14 34495, ax-c15 34493, and ax-c16 34496: in that system, we can derive any instance of ax-5 1879 not containing wff variables by induction on formula length, using ax5eq 34536 and ax5el 34541 for the basis together with hbn 2184, hbal 2076, and hbim 2165. However, if we omit this axiom, our development would be quite inconvenient since we could work only with specific instances of wffs containing no wff variables - this axiom introduces the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). (Contributed by NM, 19-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜑 → ∀𝑥𝜑)

Theoremsps-o 34512 Generalization of antecedent. (Contributed by NM, 5-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theoremhbequid 34513 Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof does not use ax-c10 34490.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)

Theoremnfequid-o 34514 Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof uses only ax-4 1777, ax-7 1981, ax-c9 34494, and ax-gen 1762. This shows that this can be proved without ax6 2287, even though the theorem equid 1985 cannot be. A shorter proof using ax6 2287 is obtainable from equid 1985 and hbth 1769.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1946, which is used for the derivation of axc9 2338, unless we consider ax-c9 34494 the starting axiom rather than ax-13 2282. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥 = 𝑥

Theoremaxc5c7 34515 Proof of a single axiom that can replace ax-c5 34487 and ax-c7 34489. See axc5c7toc5 34516 and axc5c7toc7 34517 for the rederivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) → 𝜑)

Theoremaxc5c7toc5 34516 Rederivation of ax-c5 34487 from axc5c7 34515. Only propositional calculus is used for the rederivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)

Theoremaxc5c7toc7 34517 Rederivation of ax-c7 34489 from axc5c7 34515. Only propositional calculus is used for the rederivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc711 34518 Proof of a single axiom that can replace both ax-c7 34489 and ax-11 2074. See axc711toc7 34520 and axc711to11 34521 for the rederivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑦𝑥𝜑 → ∀𝑦𝜑)

Theoremnfa1-o 34519 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑥𝜑

Theoremaxc711toc7 34520 Rederivation of ax-c7 34489 from axc711 34518. Note that ax-c7 34489 and ax-11 2074 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc711to11 34521 Rederivation of ax-11 2074 from axc711 34518. Note that ax-c7 34489 and ax-11 2074 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremaxc5c711 34522 Proof of a single axiom that can replace ax-c5 34487, ax-c7 34489, and ax-11 2074 in a subsystem that includes these axioms plus ax-c4 34488 and ax-gen 1762 (and propositional calculus). See axc5c711toc5 34523, axc5c711toc7 34524, and axc5c711to11 34525 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 34515. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑) → 𝜑)

Theoremaxc5c711toc5 34523 Rederivation of ax-c5 34487 from axc5c711 34522. Only propositional calculus is used by the rederivation. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)

Theoremaxc5c711toc7 34524 Rederivation of ax-c7 34489 from axc5c711 34522. Note that ax-c7 34489 and ax-11 2074 are not used by the rederivation. The use of alimi 1779 (which uses ax-c5 34487) is allowed since we have already proved axc5c711toc5 34523. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc5c711to11 34525 Rederivation of ax-11 2074 from axc5c711 34522. Note that ax-c7 34489 and ax-11 2074 are not used by the rederivation. The use of alimi 1779 (which uses ax-c5 34487) is allowed since we have already proved axc5c711toc5 34523. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremequidqe 34526 equid 1985 with existential quantifier without using ax-c5 34487 or ax-5 1879. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑦 ¬ 𝑥 = 𝑥

Theoremaxc5sp1 34527 A special case of ax-c5 34487 without using ax-c5 34487 or ax-5 1879. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)

Theoremequidq 34528 equid 1985 with universal quantifier without using ax-c5 34487 or ax-5 1879. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥 = 𝑥

Theoremequid1ALT 34529 Alternate proof of equid 1985 and equid1 34503 from older axioms ax-c7 34489, ax-c10 34490 and ax-c9 34494. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑥

Theoremaxc11nfromc11 34530 Rederivation of ax-c11n 34492 from original version ax-c11 34491. See theorem axc11 2347 for the derivation of ax-c11 34491 from ax-c11n 34492.

This theorem should not be referenced in any proof. Instead, use ax-c11n 34492 above so that uses of ax-c11n 34492 can be more easily identified, or use aecom-o 34505 when this form is needed for studies involving ax-c11 34491 and omitting ax-5 1879. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremnaecoms-o 34531 A commutation rule for distinct variable specifiers. Version of naecoms 2346 using ax-c11 34491. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑦 𝑦 = 𝑥𝜑)

Theoremhbnae-o 34532 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 2350 using ax-c11 34491. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremdvelimf-o 34533 Proof of dvelimh 2367 that uses ax-c11 34491 but not ax-c15 34493, ax-c11n 34492, or ax-12 2087. Version of dvelimh 2367 using ax-c11 34491 instead of axc11 2347. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremdral2-o 34534 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral2 2355 using ax-c11 34491. (Contributed by NM, 27-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))

Theoremaev-o 34535* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-c16 34496. Version of aev 2025 using ax-c11 34491. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)

Theoremax5eq 34536* Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 1879 considered as a metatheorem. Do not use it for later proofs - use ax-5 1879 instead, to avoid reference to the redundant axiom ax-c16 34496.) (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)

Theoremdveeq2-o 34537* Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2334 using ax-c15 34493. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremaxc16g-o 34538* A generalization of axiom ax-c16 34496. Version of axc16g 2172 using ax-c11 34491. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Theoremdveeq1-o 34539* Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2336 using ax-c11 . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremdveeq1-o16 34540* Version of dveeq1 2336 using ax-c16 34496 instead of ax-5 1879. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1879. (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremax5el 34541* Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 1879 considered as a metatheorem.) (Contributed by NM, 22-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥𝑦 → ∀𝑧 𝑥𝑦)

Theoremaxc11n-16 34542* This theorem shows that, given ax-c16 34496, we can derive a version of ax-c11n 34492. However, it is weaker than ax-c11n 34492 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)

Theoremdveel2ALT 34543* Alternate proof of dveel2 2399 using ax-c16 34496 instead of ax-5 1879. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))

Theoremax12f 34544 Basis step for constructing a substitution instance of ax-c15 34493 without using ax-c15 34493. We can start with any formula 𝜑 in which 𝑥 is not free. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax12eq 34545 Basis step for constructing a substitution instance of ax-c15 34493 without using ax-c15 34493. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦𝑧 = 𝑤))))

Theoremax12el 34546 Basis step for constructing a substitution instance of ax-c15 34493 without using ax-c15 34493. Atomic formula for membership predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝑧𝑤 → ∀𝑥(𝑥 = 𝑦𝑧𝑤))))

Theoremax12indn 34547 Induction step for constructing a substitution instance of ax-c15 34493 without using ax-c15 34493. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))))

Theoremax12indi 34548 Induction step for constructing a substitution instance of ax-c15 34493 without using ax-c15 34493. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))    &   (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜓 → ∀𝑥(𝑥 = 𝑦𝜓))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → ((𝜑𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))))

Theoremax12indalem 34549 Lemma for ax12inda2 34551 and ax12inda 34552. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))

Theoremax12inda2ALT 34550* Alternate proof of ax12inda2 34551, slightly more direct and not requiring ax-c16 34496. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))

Theoremax12inda2 34551* Induction step for constructing a substitution instance of ax-c15 34493 without using ax-c15 34493. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 34552. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))

Theoremax12inda 34552* Induction step for constructing a substitution instance of ax-c15 34493 without using ax-c15 34493. Quantification case. (When 𝑧 and 𝑦 are distinct, ax12inda2 34551 may be used instead to avoid the dummy variable 𝑤 in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑤 → (𝑥 = 𝑤 → (𝜑 → ∀𝑥(𝑥 = 𝑤𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))

Theoremax12v2-o 34553* Rederivation of ax-c15 34493 from ax12v 2088 (without using ax-c15 34493 or the full ax-12 2087). Thus, the hypothesis (ax12v 2088) provides an alternate axiom that can be used in place of ax-c15 34493. See also axc15 2339. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax12a2-o 34554* Derive ax-c15 34493 from a hypothesis in the form of ax-12 2087, without using ax-12 2087 or ax-c15 34493. The hypothesis is weaker than ax-12 2087, with 𝑧 both distinct from 𝑥 and not occurring in 𝜑. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-12 2087, if we also have ax-c11 34491, which this proof uses. As theorem ax12 2340 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 34492 instead of ax-c11 34491. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremaxc11-o 34555 Show that ax-c11 34491 can be derived from ax-c11n 34492 and ax-12 2087. An open problem is whether this theorem can be derived from ax-c11n 34492 and the others when ax-12 2087 is replaced with ax-c15 34493 or ax12v 2088. See theorem axc11nfromc11 34530 for the rederivation of ax-c11n 34492 from axc11 2347.

Normally, axc11 2347 should be used rather than ax-c11 34491 or axc11-o 34555, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theoremfsumshftd 34556* Index shift of a finite sum with a weaker "implicit substitution" hypothesis than fsumshft 14556. The proof demonstrates how this can be derived starting from from fsumshft 14556. (Contributed by NM, 1-Nov-2019.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ((𝜑𝑗 = (𝑘𝐾)) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵)

Axiomax-riotaBAD 34557 Define restricted description binder. In case it doesn't exist, we return a set which is not a member of the domain of discourse 𝐴. See also comments for df-iota 5889. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) WARNING: THIS "AXIOM", WHICH IS THE OLD df-riota 6651, CONFLICTS WITH (THE NEW) df-riota 6651 AND MAKES THE SYSTEM IN set.mm INCONSISTENT. IT IS TEMPORARY AND WILL BE DELETED AFTER ALL USES ARE ELIMINATED.
(𝑥𝐴 𝜑) = if(∃!𝑥𝐴 𝜑, (℩𝑥(𝑥𝐴𝜑)), (Undef‘{𝑥𝑥𝐴}))

TheoremriotaclbgBAD 34558* Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝐴𝑉 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))

TheoremriotaclbBAD 34559* Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
𝐴 ∈ V       (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴)

Theoremriotasvd 34560* Deduction version of riotasv 34563. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   (𝜑𝐷𝐴)       ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))

Theoremriotasv2d 34561* Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4904). Special case of riota2f 6672. (Contributed by NM, 2-Mar-2013.)
𝑦𝜑    &   (𝜑𝑦𝐹)    &   (𝜑 → Ⅎ𝑦𝜒)    &   (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   ((𝜑𝑦 = 𝐸) → (𝜓𝜒))    &   ((𝜑𝑦 = 𝐸) → 𝐶 = 𝐹)    &   (𝜑𝐷𝐴)    &   (𝜑𝐸𝐵)    &   (𝜑𝜒)       ((𝜑𝐴𝑉) → 𝐷 = 𝐹)

Theoremriotasv2s 34562* The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4904) in the form of a substitution instance. Special case of riota2f 6672. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))       ((𝐴𝑉𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐷 = 𝐸 / 𝑦𝐶)

Theoremriotasv 34563* Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4904). Special case of riota2f 6672. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
𝐴 ∈ V    &   𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))       ((𝐷𝐴𝑦𝐵𝜑) → 𝐷 = 𝐶)

Theoremriotasv3d 34564* A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 4904) also holds for its description binder 𝐷 (in the form of property 𝜃). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜃)    &   (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   ((𝜑𝐶 = 𝐷) → (𝜒𝜃))    &   (𝜑 → ((𝑦𝐵𝜓) → 𝜒))    &   (𝜑𝐷𝐴)    &   (𝜑 → ∃𝑦𝐵 𝜓)       ((𝜑𝐴𝑉) → 𝜃)

20.23.4  Experiments with weak deduction theorem

Theoremelimhyps 34565 A version of elimhyp 4179 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
[𝐵 / 𝑥]𝜑       [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑

Theoremdedths 34566 A version of weak deduction theorem dedth 4172 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓       (𝜑𝜓)

TheoremrenegclALT 34567 Closure law for negative of reals. Demonstrates use of weak deduction theorem with explicit substitution. The proof is much longer than that of renegcl 10382. (Contributed by NM, 15-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ ℝ → -𝐴 ∈ ℝ)

Theoremelimhyps2 34568 Generalization of elimhyps 34565 that is not useful unless we can separately prove 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.)
[𝐵 / 𝑥]𝜑       [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑

Theoremdedths2 34569 Generalization of dedths 34566 that is not useful unless we can separately prove 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.)
[if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)

Theorem19.9alt 34570 Version of 19.9t 2109 for universal quantifier. (Contributed by NM, 9-Nov-2020.)
(Ⅎ𝑥𝜑 → (∀𝑥𝜑𝜑))

Theoremnfcxfrdf 34571 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by NM, 19-Nov-2020.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴)

Theoremnfded 34572 A deduction theorem that converts a not-free inference directly to deduction form. The first hypothesis is the hypothesis of the deduction form. The second is an equality deduction (e.g. (𝑥𝐴 {𝑦 ∣ ∀𝑥𝑦𝐴} = 𝐴)) that starts from abidnf 3408. The last is assigned to the inference form (e.g. 𝑥 {𝑦 ∣ ∀𝑥𝑦𝐴}) whose hypothesis is satisfied using nfaba1 2799. (Contributed by NM, 19-Nov-2020.)
(𝜑𝑥𝐴)    &   (𝑥𝐴𝐵 = 𝐶)    &   𝑥𝐵       (𝜑𝑥𝐶)

Theoremnfded2 34573 A deduction theorem that converts a not-free inference directly to deduction form. The first 2 hypotheses are the hypotheses of the deduction form. The third is an equality deduction (e.g. ((𝑥𝐴𝑥𝐵) → ⟨{𝑦 ∣ ∀𝑥𝑦𝐴}, {𝑦 ∣ ∀𝑥𝑦𝐵}⟩ = ⟨𝐴, 𝐵⟩) for nfopd 4450) that starts from abidnf 3408. The last is assigned to the inference form (e.g. 𝑥⟨{𝑦 ∣ ∀𝑥𝑦𝐴}, {𝑦 ∣ ∀𝑥𝑦𝐵}⟩ for nfop 4449) whose hypotheses are satisfied using nfaba1 2799. (Contributed by NM, 19-Nov-2020.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)    &   ((𝑥𝐴𝑥𝐵) → 𝐶 = 𝐷)    &   𝑥𝐶       (𝜑𝑥𝐷)

TheoremnfunidALT2 34574 Deduction version of nfuni 4474. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)

TheoremnfunidALT 34575 Deduction version of nfuni 4474. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)

TheoremnfopdALT 34576 Deduction version of bound-variable hypothesis builder nfop 4449. This shows how the deduction version of a not-free theorem such as nfop 4449 can be created from the corresponding not-free inference theorem. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴, 𝐵⟩)

20.23.5  Miscellanea

Theoremcnaddcom 34577 Recover the commutative law of addition for complex numbers from the Abelian group structure. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremtoycom 34578* Show the commutative law for an operation 𝑂 on a toy structure class 𝐶 of commuatitive operations on . This illustrates how a structure class can be partially specialized. In practice, we would ordinarily define a new constant such as "CAbel" in place of 𝐶. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)
𝐶 = {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ}    &    + = (+g𝐾)       ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

20.23.6  Atoms, hyperplanes, and covering in a left vector space (or module)

Syntaxclsa 34579 Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
class LSAtoms

Syntaxclsh 34580 Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
class LSHyp

Definitiondf-lsatoms 34581* Define the set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.)
LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))

Definitiondf-lshyp 34582* Define the set of all hyperplanes of a left module or left vector space. Also called co-atoms, these are subspaces that are one dimension less that the full space. (Contributed by NM, 29-Jun-2014.)
LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})

Theoremlshpset 34583* The set of all hyperplanes of a left module or left vector space. The vector 𝑣 is called a generating vector for the hyperplane. (Contributed by NM, 29-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐻 = (LSHyp‘𝑊)       (𝑊𝑋𝐻 = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})

Theoremislshp 34584* The predicate "is a hyperplane" (of a left module or left vector space). (Contributed by NM, 29-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐻 = (LSHyp‘𝑊)       (𝑊𝑋 → (𝑈𝐻 ↔ (𝑈𝑆𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))

Theoremislshpsm 34585* Hyperplane properties expressed with subspace sum. (Contributed by NM, 3-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)       (𝜑 → (𝑈𝐻 ↔ (𝑈𝑆𝑈𝑉 ∧ ∃𝑣𝑉 (𝑈 (𝑁‘{𝑣})) = 𝑉)))

Theoremlshplss 34586 A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)       (𝜑𝑈𝑆)

Theoremlshpne 34587 A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)       (𝜑𝑈𝑉)

Theoremlshpnel 34588 A hyperplane's generating vector does not belong to the hyperplane. (Contributed by NM, 3-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑋})) = 𝑉)       (𝜑 → ¬ 𝑋𝑈)

Theoremlshpnelb 34589 The subspace sum of a hyperplane and the span of an element equals the vector space iff the element is not in the hyperplane. (Contributed by NM, 2-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑋𝑉)       (𝜑 → (¬ 𝑋𝑈 ↔ (𝑈 (𝑁‘{𝑋})) = 𝑉))

Theoremlshpnel2N 34590 Condition that determines a hyperplane. (Contributed by NM, 3-Oct-2014.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋𝑈)       (𝜑 → (𝑈𝐻 ↔ (𝑈 (𝑁‘{𝑋})) = 𝑉))

Theoremlshpne0 34591 The member of the span in the hyperplane definition does not belong to the hyperplane. (Contributed by NM, 14-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑋})) = 𝑉)       (𝜑𝑋0 )

Theoremlshpdisj 34592 A hyperplane and the span in the hyperplane definition are disjoint. (Contributed by NM, 3-Jul-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑋})) = 𝑉)       (𝜑 → (𝑈 ∩ (𝑁‘{𝑋})) = { 0 })

Theoremlshpcmp 34593 If two hyperplanes are comparable, they are equal. (Contributed by NM, 9-Oct-2014.)
𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐻)    &   (𝜑𝑈𝐻)       (𝜑 → (𝑇𝑈𝑇 = 𝑈))

TheoremlshpinN 34594 The intersection of two different hyperplanes is not a hyperplane. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐻)    &   (𝜑𝑈𝐻)       (𝜑 → ((𝑇𝑈) ∈ 𝐻𝑇 = 𝑈))

Theoremlsatset 34595* The set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)       (𝑊𝑋𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))

Theoremislsat 34596* The predicate "is a 1-dim subspace (atom)" (of a left module or left vector space). (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)       (𝑊𝑋 → (𝑈𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥})))

Theoremlsatlspsn2 34597 The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 34598 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → (𝑁‘{𝑋}) ∈ 𝐴)

Theoremlsatlspsn 34598 The span of a nonzero singleton is an atom. (Contributed by NM, 16-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴)

Theoremislsati 34599* A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       ((𝑊𝑋𝑈𝐴) → ∃𝑣𝑉 𝑈 = (𝑁‘{𝑣}))

Theoremlsateln0 34600* A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐴)       (𝜑 → ∃𝑣𝑈 𝑣0 )

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