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Theorem unisnALT 38982
Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. The User manually input on a mmj2 Proof Worksheet, without labels, all steps of unisnALT 38982 except 1, 11, 15, 21, and 30. With execution of the mmj2 unification command, mmj2 could find labels for all steps except for 2, 12, 16, 22, and 31 (and the then non-existing steps 1, 11, 15, 21, and 30) . mmj2 could not find reference theorems for those five steps because the hypothesis field of each of these steps was empty and none of those steps unifies with a theorem in set.mm. Each of these five steps is a semantic variation of a theorem in set.mm and is 2-step provable. mmj2 does not have the ability to automatically generate the semantic variation in set.mm of a theorem in a mmj2 Proof Worksheet unless the theorem in the Proof Worksheet is labeled with a 1-hypothesis deduction whose hypothesis is a theorem in set.mm which unifies with the theorem in the Proof Worksheet. The stepprover.c program, which invokes mmj2, has this capability. stepprover.c automatically generated steps 1, 11, 15, 21, and 30, labeled all steps, and generated the RPN proof of unisnALT 38982. Roughly speaking, stepprover.c added to the Proof Worksheet a labeled duplicate step of each non-unifying theorem for each label in a text file, labels.txt, containing a list of labels provided by the User. Upon mmj2 unification, stepprover.c identified a label for each of the five theorems which 2-step proves it. For unisnALT 38982, the label list is a list of all 1-hypothesis propositional calculus deductions in set.mm. stepproverp.c is the same as stepprover.c except that it intermittently pauses during execution, allowing the User to observe the changes to a text file caused by the execution of particular statements of the program. (Contributed by Alan Sare, 19-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
unisnALT.1 𝐴 ∈ V
Assertion
Ref Expression
unisnALT {𝐴} = 𝐴

Proof of Theorem unisnALT
Dummy variables 𝑥 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 4430 . . . . . 6 (𝑥 {𝐴} ↔ ∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}))
21biimpi 206 . . . . 5 (𝑥 {𝐴} → ∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}))
3 id 22 . . . . . . . . 9 ((𝑥𝑞𝑞 ∈ {𝐴}) → (𝑥𝑞𝑞 ∈ {𝐴}))
4 simpl 473 . . . . . . . . 9 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝑞)
53, 4syl 17 . . . . . . . 8 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝑞)
6 simpr 477 . . . . . . . . . 10 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
73, 6syl 17 . . . . . . . . 9 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
8 elsni 4185 . . . . . . . . 9 (𝑞 ∈ {𝐴} → 𝑞 = 𝐴)
97, 8syl 17 . . . . . . . 8 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑞 = 𝐴)
10 eleq2 2688 . . . . . . . . 9 (𝑞 = 𝐴 → (𝑥𝑞𝑥𝐴))
1110biimpac 503 . . . . . . . 8 ((𝑥𝑞𝑞 = 𝐴) → 𝑥𝐴)
125, 9, 11syl2anc 692 . . . . . . 7 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)
1312ax-gen 1720 . . . . . 6 𝑞((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)
14 19.23v 1900 . . . . . . 7 (∀𝑞((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴) ↔ (∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴))
1514biimpi 206 . . . . . 6 (∀𝑞((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴) → (∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴))
1613, 15ax-mp 5 . . . . 5 (∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)
17 pm3.35 610 . . . . 5 ((∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) ∧ (∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)) → 𝑥𝐴)
182, 16, 17sylancl 693 . . . 4 (𝑥 {𝐴} → 𝑥𝐴)
1918ax-gen 1720 . . 3 𝑥(𝑥 {𝐴} → 𝑥𝐴)
20 dfss2 3584 . . . 4 ( {𝐴} ⊆ 𝐴 ↔ ∀𝑥(𝑥 {𝐴} → 𝑥𝐴))
2120biimpri 218 . . 3 (∀𝑥(𝑥 {𝐴} → 𝑥𝐴) → {𝐴} ⊆ 𝐴)
2219, 21ax-mp 5 . 2 {𝐴} ⊆ 𝐴
23 id 22 . . . . 5 (𝑥𝐴𝑥𝐴)
24 unisnALT.1 . . . . . 6 𝐴 ∈ V
2524snid 4199 . . . . 5 𝐴 ∈ {𝐴}
26 elunii 4432 . . . . 5 ((𝑥𝐴𝐴 ∈ {𝐴}) → 𝑥 {𝐴})
2723, 25, 26sylancl 693 . . . 4 (𝑥𝐴𝑥 {𝐴})
2827ax-gen 1720 . . 3 𝑥(𝑥𝐴𝑥 {𝐴})
29 dfss2 3584 . . . 4 (𝐴 {𝐴} ↔ ∀𝑥(𝑥𝐴𝑥 {𝐴}))
3029biimpri 218 . . 3 (∀𝑥(𝑥𝐴𝑥 {𝐴}) → 𝐴 {𝐴})
3128, 30ax-mp 5 . 2 𝐴 {𝐴}
3222, 31eqssi 3611 1 {𝐴} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1479   = wceq 1481  wex 1702  wcel 1988  Vcvv 3195  wss 3567  {csn 4168   cuni 4427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-in 3574  df-ss 3581  df-sn 4169  df-uni 4428
This theorem is referenced by: (None)
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