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Theorem unisnALT 30231
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 30231 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 30231. 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 30231, 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 4104 . . . . . 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 3919 . . . . . . . . 9
97, 8syl 16 . . . . . . . 8
10 eleq2 2542 . . . . . . . . 9
1110biimpac 474 . . . . . . . 8
125, 9, 11syl2anc 644 . . . . . . 7
1312ax-gen 1562 . . . . . 6
14 19.23v 1910 . . . . . . 7
1514biimpi 188 . . . . . 6
1613, 15ax-mp 5 . . . . 5
17 pm3.35 572 . . . . 5
182, 16, 17sylancl 645 . . . 4
1918ax-gen 1562 . . 3
20 dfss2 3370 . . . 4
2120biimpri 199 . . 3
2219, 21ax-mp 5 . 2
23 id 21 . . . . 5
24 unisnALT.1 . . . . . 6
2524snid 3922 . . . . 5
26 elunii 4106 . . . . 5
2723, 25, 26sylancl 645 . . . 4
2827ax-gen 1562 . . 3
29 dfss2 3370 . . . 4
3029biimpri 199 . . 3
3128, 30ax-mp 5 . 2
3222, 31eqssi 3397 1
Colors of variables: wff set class
Syntax hints:  ->wi 4  /\wa 360  A.wal 1556  E.wex 1557  =wceq 1662  e.wcel 1724   cvv 3006  C_wss 3353  {csn 3894  U.cuni 4101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1562  ax-4 1573  ax-5 1636  ax-6 1677  ax-7 1697  ax-10 1743  ax-11 1748  ax-12 1760  ax-13 1947  ax-ext 2462
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1337  df-ex 1558  df-nf 1561  df-sb 1669  df-clab 2468  df-cleq 2474  df-clel 2477  df-nfc 2606  df-v 3008  df-in 3360  df-ss 3367  df-sn 3900  df-uni 4102
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