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Theorem unisnALT 42027
 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 42027 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 42027. 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 42027, 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 4801 . . . . . 6 (𝑥 {𝐴} ↔ ∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}))
21biimpi 219 . . . . 5 (𝑥 {𝐴} → ∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}))
3 id 22 . . . . . . . . 9 ((𝑥𝑞𝑞 ∈ {𝐴}) → (𝑥𝑞𝑞 ∈ {𝐴}))
4 simpl 486 . . . . . . . . 9 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝑞)
53, 4syl 17 . . . . . . . 8 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝑞)
6 simpr 488 . . . . . . . . . 10 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
73, 6syl 17 . . . . . . . . 9 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
8 elsni 4539 . . . . . . . . 9 (𝑞 ∈ {𝐴} → 𝑞 = 𝐴)
97, 8syl 17 . . . . . . . 8 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑞 = 𝐴)
10 eleq2 2840 . . . . . . . . 9 (𝑞 = 𝐴 → (𝑥𝑞𝑥𝐴))
1110biimpac 482 . . . . . . . 8 ((𝑥𝑞𝑞 = 𝐴) → 𝑥𝐴)
125, 9, 11syl2anc 587 . . . . . . 7 ((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)
1312ax-gen 1797 . . . . . 6 𝑞((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)
14 19.23v 1943 . . . . . . 7 (∀𝑞((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴) ↔ (∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴))
1514biimpi 219 . . . . . 6 (∀𝑞((𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴) → (∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴))
1613, 15ax-mp 5 . . . . 5 (∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)
17 pm3.35 802 . . . . 5 ((∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) ∧ (∃𝑞(𝑥𝑞𝑞 ∈ {𝐴}) → 𝑥𝐴)) → 𝑥𝐴)
182, 16, 17sylancl 589 . . . 4 (𝑥 {𝐴} → 𝑥𝐴)
1918ax-gen 1797 . . 3 𝑥(𝑥 {𝐴} → 𝑥𝐴)
20 dfss2 3878 . . . 4 ( {𝐴} ⊆ 𝐴 ↔ ∀𝑥(𝑥 {𝐴} → 𝑥𝐴))
2120biimpri 231 . . 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 4558 . . . . 5 𝐴 ∈ {𝐴}
26 elunii 4803 . . . . 5 ((𝑥𝐴𝐴 ∈ {𝐴}) → 𝑥 {𝐴})
2723, 25, 26sylancl 589 . . . 4 (𝑥𝐴𝑥 {𝐴})
2827ax-gen 1797 . . 3 𝑥(𝑥𝐴𝑥 {𝐴})
29 dfss2 3878 . . . 4 (𝐴 {𝐴} ↔ ∀𝑥(𝑥𝐴𝑥 {𝐴}))
3029biimpri 231 . . 3 (∀𝑥(𝑥𝐴𝑥 {𝐴}) → 𝐴 {𝐴})
3128, 30ax-mp 5 . 2 𝐴 {𝐴}
3222, 31eqssi 3908 1 {𝐴} = 𝐴
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3858  {csn 4522  ∪ cuni 4798 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3865  df-ss 3875  df-sn 4523  df-uni 4799 This theorem is referenced by: (None)
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