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Theorem unisnALT 43458
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 43458 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 43458. 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 43458, 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 4904 . . . . . 6 (𝑥 {𝐴} ↔ ∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}))
21biimpi 215 . . . . 5 (𝑥 {𝐴} → ∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}))
3 id 22 . . . . . . . . 9 ((𝑥𝑞𝑞 ∈ {𝐴}) → (𝑥𝑞𝑞 ∈ {𝐴}))
4 simpl 483 . . . . . . . . 9 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝑞)
53, 4syl 17 . . . . . . . 8 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝑞)
6 simpr 485 . . . . . . . . . 10 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
73, 6syl 17 . . . . . . . . 9 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
8 elsni 4639 . . . . . . . . 9 (𝑞 ∈ {𝐴} → 𝑞 = 𝐴)
97, 8syl 17 . . . . . . . 8 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑞 = 𝐴)
10 eleq2 2821 . . . . . . . . 9 (𝑞 = 𝐴 → (𝑥𝑞𝑥𝐴))
1110biimpac 479 . . . . . . . 8 ((𝑥𝑞𝑞 = 𝐴) → 𝑥𝐴)
125, 9, 11syl2anc 584 . . . . . . 7 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)
1312ax-gen 1797 . . . . . 6 𝑞((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)
14 19.23v 1945 . . . . . . 7 (∀𝑞((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴) ↔ (∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴))
1514biimpi 215 . . . . . 6 (∀𝑞((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴) → (∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴))
1613, 15ax-mp 5 . . . . 5 (∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)
17 pm3.35 801 . . . . 5 ((∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) ∧ (∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)) → 𝑥𝐴)
182, 16, 17sylancl 586 . . . 4 (𝑥 {𝐴} → 𝑥𝐴)
1918ax-gen 1797 . . 3 𝑥(𝑥 {𝐴} → 𝑥𝐴)
20 dfss2 3964 . . . 4 ( {𝐴} ⊆ 𝐴 ↔ ∀𝑥(𝑥 {𝐴} → 𝑥𝐴))
2120biimpri 227 . . 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 4658 . . . . 5 𝐴 ∈ {𝐴}
26 elunii 4906 . . . . 5 ((𝑥𝐴𝐴 ∈ {𝐴}) → 𝑥 {𝐴})
2723, 25, 26sylancl 586 . . . 4 (𝑥𝐴𝑥 {𝐴})
2827ax-gen 1797 . . 3 𝑥(𝑥𝐴𝑥 {𝐴})
29 dfss2 3964 . . . 4 (𝐴 {𝐴} ↔ ∀𝑥(𝑥𝐴𝑥 {𝐴}))
3029biimpri 227 . . 3 (∀𝑥(𝑥𝐴𝑥 {𝐴}) → 𝐴 {𝐴})
3128, 30ax-mp 5 . 2 𝐴 {𝐴}
3222, 31eqssi 3994 1 {𝐴} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  Vcvv 3473  wss 3944  {csn 4622   cuni 4901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3951  df-ss 3961  df-sn 4623  df-uni 4902
This theorem is referenced by: (None)
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