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Theorem unisnALT 31662
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 31662 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 31662. 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 31662, 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  e. 
_V
Assertion
Ref Expression
unisnALT  |-  U. { A }  =  A

Proof of Theorem unisnALT
Dummy variables  x  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 4094 . . . . . 6  |-  ( x  e.  U. { A } 
<->  E. q ( x  e.  q  /\  q  e.  { A } ) )
21biimpi 194 . . . . 5  |-  ( x  e.  U. { A }  ->  E. q ( x  e.  q  /\  q  e.  { A } ) )
3 id 22 . . . . . . . . 9  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  (
x  e.  q  /\  q  e.  { A } ) )
4 simpl 457 . . . . . . . . 9  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  q )
53, 4syl 16 . . . . . . . 8  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  q )
6 simpr 461 . . . . . . . . . 10  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  q  e.  { A } )
73, 6syl 16 . . . . . . . . 9  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  q  e.  { A } )
8 elsni 3902 . . . . . . . . 9  |-  ( q  e.  { A }  ->  q  =  A )
97, 8syl 16 . . . . . . . 8  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  q  =  A )
10 eleq2 2504 . . . . . . . . 9  |-  ( q  =  A  ->  (
x  e.  q  <->  x  e.  A ) )
1110biimpac 486 . . . . . . . 8  |-  ( ( x  e.  q  /\  q  =  A )  ->  x  e.  A )
125, 9, 11syl2anc 661 . . . . . . 7  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A )
1312ax-gen 1591 . . . . . 6  |-  A. q
( ( x  e.  q  /\  q  e. 
{ A } )  ->  x  e.  A
)
14 19.23v 1910 . . . . . . 7  |-  ( A. q ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A
)  <->  ( E. q
( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A ) )
1514biimpi 194 . . . . . 6  |-  ( A. q ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A
)  ->  ( E. q ( x  e.  q  /\  q  e. 
{ A } )  ->  x  e.  A
) )
1613, 15ax-mp 5 . . . . 5  |-  ( E. q ( x  e.  q  /\  q  e. 
{ A } )  ->  x  e.  A
)
17 pm3.35 587 . . . . 5  |-  ( ( E. q ( x  e.  q  /\  q  e.  { A } )  /\  ( E. q
( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A ) )  ->  x  e.  A )
182, 16, 17sylancl 662 . . . 4  |-  ( x  e.  U. { A }  ->  x  e.  A
)
1918ax-gen 1591 . . 3  |-  A. x
( x  e.  U. { A }  ->  x  e.  A )
20 dfss2 3345 . . . 4  |-  ( U. { A }  C_  A  <->  A. x ( x  e. 
U. { A }  ->  x  e.  A ) )
2120biimpri 206 . . 3  |-  ( A. x ( x  e. 
U. { A }  ->  x  e.  A )  ->  U. { A }  C_  A )
2219, 21ax-mp 5 . 2  |-  U. { A }  C_  A
23 id 22 . . . . 5  |-  ( x  e.  A  ->  x  e.  A )
24 unisnALT.1 . . . . . 6  |-  A  e. 
_V
2524snid 3905 . . . . 5  |-  A  e. 
{ A }
26 elunii 4096 . . . . 5  |-  ( ( x  e.  A  /\  A  e.  { A } )  ->  x  e.  U. { A }
)
2723, 25, 26sylancl 662 . . . 4  |-  ( x  e.  A  ->  x  e.  U. { A }
)
2827ax-gen 1591 . . 3  |-  A. x
( x  e.  A  ->  x  e.  U. { A } )
29 dfss2 3345 . . . 4  |-  ( A 
C_  U. { A }  <->  A. x ( x  e.  A  ->  x  e.  U. { A } ) )
3029biimpri 206 . . 3  |-  ( A. x ( x  e.  A  ->  x  e.  U. { A } )  ->  A  C_  U. { A } )
3128, 30ax-mp 5 . 2  |-  A  C_  U. { A }
3222, 31eqssi 3372 1  |-  U. { A }  =  A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2972    C_ wss 3328   {csn 3877   U.cuni 4091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2974  df-in 3335  df-ss 3342  df-sn 3878  df-uni 4092
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator