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Theorem unisnALT 36737
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 36737 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 36737. 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 36737, 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 4193 . . . . . 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 455 . . . . . . . . 9  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  q )
53, 4syl 17 . . . . . . . 8  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  q )
6 simpr 459 . . . . . . . . . 10  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  q  e.  { A } )
73, 6syl 17 . . . . . . . . 9  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  q  e.  { A } )
8 elsni 3996 . . . . . . . . 9  |-  ( q  e.  { A }  ->  q  =  A )
97, 8syl 17 . . . . . . . 8  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  q  =  A )
10 eleq2 2475 . . . . . . . . 9  |-  ( q  =  A  ->  (
x  e.  q  <->  x  e.  A ) )
1110biimpac 484 . . . . . . . 8  |-  ( ( x  e.  q  /\  q  =  A )  ->  x  e.  A )
125, 9, 11syl2anc 659 . . . . . . 7  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A )
1312ax-gen 1639 . . . . . 6  |-  A. q
( ( x  e.  q  /\  q  e. 
{ A } )  ->  x  e.  A
)
14 19.23v 1784 . . . . . . 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 pm3.35 585 . . . . 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 660 . . . 4  |-  ( x  e.  U. { A }  ->  x  e.  A
)
1918ax-gen 1639 . . 3  |-  A. x
( x  e.  U. { A }  ->  x  e.  A )
20 dfss2 3430 . . . 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 3999 . . . . 5  |-  A  e. 
{ A }
26 elunii 4195 . . . . 5  |-  ( ( x  e.  A  /\  A  e.  { A } )  ->  x  e.  U. { A }
)
2723, 25, 26sylancl 660 . . . 4  |-  ( x  e.  A  ->  x  e.  U. { A }
)
2827ax-gen 1639 . . 3  |-  A. x
( x  e.  A  ->  x  e.  U. { A } )
29 dfss2 3430 . . . 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 3457 1  |-  U. { A }  =  A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1403    = wceq 1405   E.wex 1633    e. wcel 1842   _Vcvv 3058    C_ wss 3413   {csn 3971   U.cuni 4190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-in 3420  df-ss 3427  df-sn 3972  df-uni 4191
This theorem is referenced by: (None)
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