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Theorem unisnALT 29435
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 29435 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 29435. 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 29435, 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
Assertion
Ref Expression
unisnALT

Proof of Theorem unisnALT
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 4050 . . . . . 6
21biimpi 188 . . . . 5
3 id 21 . . . . . . . . 9
4 simpl 445 . . . . . . . . 9
53, 4syl 16 . . . . . . . 8
6 simpr 449 . . . . . . . . . 10
73, 6syl 16 . . . . . . . . 9
8 elsni 3869 . . . . . . . . 9
97, 8syl 16 . . . . . . . 8
10 eleq2 2508 . . . . . . . . 9
1110biimpac 474 . . . . . . . 8
125, 9, 11syl2anc 644 . . . . . . 7
1312ax-gen 1556 . . . . . 6
14 19.23v 1919 . . . . . . 7
1514biimpi 188 . . . . . 6
1613, 15ax-mp 5 . . . . 5
17 pm3.35 572 . . . . 5
182, 16, 17sylancl 645 . . . 4
1918ax-gen 1556 . . 3
20 dfss2 3330 . . . 4
2120biimpri 199 . . 3
2219, 21ax-mp 5 . 2
23 id 21 . . . . 5
24 unisnALT.1 . . . . . 6
2524snid 3872 . . . . 5
26 elunii 4052 . . . . 5
2723, 25, 26sylancl 645 . . . 4
2827ax-gen 1556 . . 3
29 dfss2 3330 . . . 4
3029biimpri 199 . . 3
3128, 30ax-mp 5 . 2
3222, 31eqssi 3357 1
Colors of variables: wff set class
Syntax hints:  ->wi 4  /\wa 360  A.wal 1550  E.wex 1551  =wceq 1654  e.wcel 1728   cvv 2969  C_wss 3313  {csn 3845  U.cuni 4047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1955  ax-ext 2428
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2434  df-cleq 2440  df-clel 2443  df-nfc 2572  df-v 2971  df-in 3320  df-ss 3327  df-sn 3851  df-uni 4048
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