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Theorem unisnALT 39678
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 39678 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 39678. 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 39678, 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 4575 . . . . . 6 (𝑥 {𝐴} ↔ ∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}))
21biimpi 206 . . . . 5 (𝑥 {𝐴} → ∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}))
3 id 22 . . . . . . . . 9 ((𝑥𝑞𝑞 ∈ {𝐴}) → (𝑥𝑞𝑞 ∈ {𝐴}))
4 simpl 468 . . . . . . . . 9 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝑞)
53, 4syl 17 . . . . . . . 8 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝑞)
6 simpr 471 . . . . . . . . . 10 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
73, 6syl 17 . . . . . . . . 9 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
8 elsni 4331 . . . . . . . . 9 (𝑞 ∈ {𝐴} → 𝑞 = 𝐴)
97, 8syl 17 . . . . . . . 8 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑞 = 𝐴)
10 eleq2 2838 . . . . . . . . 9 (𝑞 = 𝐴 → (𝑥𝑞𝑥𝐴))
1110biimpac 464 . . . . . . . 8 ((𝑥𝑞𝑞 = 𝐴) → 𝑥𝐴)
125, 9, 11syl2anc 565 . . . . . . 7 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)
1312ax-gen 1869 . . . . . 6 𝑞((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)
14 19.23v 2022 . . . . . . 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 pm3.35 810 . . . . 5 ((∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) ∧ (∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)) → 𝑥𝐴)
182, 16, 17sylancl 566 . . . 4 (𝑥 {𝐴} → 𝑥𝐴)
1918ax-gen 1869 . . 3 𝑥(𝑥 {𝐴} → 𝑥𝐴)
20 dfss2 3738 . . . 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 4345 . . . . 5 𝐴 ∈ {𝐴}
26 elunii 4577 . . . . 5 ((𝑥𝐴𝐴 ∈ {𝐴}) → 𝑥 {𝐴})
2723, 25, 26sylancl 566 . . . 4 (𝑥𝐴𝑥 {𝐴})
2827ax-gen 1869 . . 3 𝑥(𝑥𝐴𝑥 {𝐴})
29 dfss2 3738 . . . 4 (𝐴 {𝐴} ↔ ∀𝑥(𝑥𝐴𝑥 {𝐴}))
3029biimpri 218 . . 3 (∀𝑥(𝑥𝐴𝑥 {𝐴}) → 𝐴 {𝐴})
3128, 30ax-mp 5 . 2 𝐴 {𝐴}
3222, 31eqssi 3766 1 {𝐴} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1628   = wceq 1630  wex 1851  wcel 2144  Vcvv 3349  wss 3721  {csn 4314   cuni 4572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-in 3728  df-ss 3735  df-sn 4315  df-uni 4573
This theorem is referenced by: (None)
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