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Theorem unisnALT 27982
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 27982 except 1, 11, 15, 21, and 30. With execution of the mmj2 unification command, mmj2 could find labels for all of 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 an 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 27982. 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 27982, 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 3831 . . . . . 6  |-  ( x  e.  U. { A } 
<->  E. q ( x  e.  q  /\  q  e.  { A } ) )
21biimpi 186 . . . . 5  |-  ( x  e.  U. { A }  ->  E. q ( x  e.  q  /\  q  e.  { A } ) )
3 id 19 . . . . . . . . 9  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  (
x  e.  q  /\  q  e.  { A } ) )
4 simpl 443 . . . . . . . . 9  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  q )
53, 4syl 15 . . . . . . . 8  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  q )
6 simpr 447 . . . . . . . . . 10  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  q  e.  { A } )
73, 6syl 15 . . . . . . . . 9  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  q  e.  { A } )
8 elsni 3665 . . . . . . . . 9  |-  ( q  e.  { A }  ->  q  =  A )
97, 8syl 15 . . . . . . . 8  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  q  =  A )
10 eleq2 2345 . . . . . . . . 9  |-  ( q  =  A  ->  (
x  e.  q  <->  x  e.  A ) )
1110biimpac 472 . . . . . . . 8  |-  ( ( x  e.  q  /\  q  =  A )  ->  x  e.  A )
125, 9, 11syl2anc 642 . . . . . . 7  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A )
1312ax-gen 1533 . . . . . 6  |-  A. q
( ( x  e.  q  /\  q  e. 
{ A } )  ->  x  e.  A
)
14 19.23v 1834 . . . . . . 7  |-  ( A. q ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A
)  <->  ( E. q
( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A ) )
1514biimpi 186 . . . . . 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 8 . . . . 5  |-  ( E. q ( x  e.  q  /\  q  e. 
{ A } )  ->  x  e.  A
)
17 pm3.35 570 . . . . 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 643 . . . 4  |-  ( x  e.  U. { A }  ->  x  e.  A
)
1918ax-gen 1533 . . 3  |-  A. x
( x  e.  U. { A }  ->  x  e.  A )
20 dfss2 3170 . . . 4  |-  ( U. { A }  C_  A  <->  A. x ( x  e. 
U. { A }  ->  x  e.  A ) )
2120biimpri 197 . . 3  |-  ( A. x ( x  e. 
U. { A }  ->  x  e.  A )  ->  U. { A }  C_  A )
2219, 21ax-mp 8 . 2  |-  U. { A }  C_  A
23 id 19 . . . . 5  |-  ( x  e.  A  ->  x  e.  A )
24 unisnALT.1 . . . . . 6  |-  A  e. 
_V
2524snid 3668 . . . . 5  |-  A  e. 
{ A }
26 elunii 3833 . . . . 5  |-  ( ( x  e.  A  /\  A  e.  { A } )  ->  x  e.  U. { A }
)
2723, 25, 26sylancl 643 . . . 4  |-  ( x  e.  A  ->  x  e.  U. { A }
)
2827ax-gen 1533 . . 3  |-  A. x
( x  e.  A  ->  x  e.  U. { A } )
29 dfss2 3170 . . . 4  |-  ( A 
C_  U. { A }  <->  A. x ( x  e.  A  ->  x  e.  U. { A } ) )
3029biimpri 197 . . 3  |-  ( A. x ( x  e.  A  ->  x  e.  U. { A } )  ->  A  C_  U. { A } )
3128, 30ax-mp 8 . 2  |-  A  C_  U. { A }
3222, 31eqssi 3196 1  |-  U. { A }  =  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1685   _Vcvv 2789    C_ wss 3153   {csn 3641   U.cuni 3828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-v 2791  df-in 3160  df-ss 3167  df-sn 3647  df-uni 3829
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