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Theorem unisnALT 27715
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 27715 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 27715. 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 27715, 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
StepHypRef Expression
1 eluni 3771 . . . . . 6  |-  ( x  e.  U. { A } 
<->  E. q ( x  e.  q  /\  q  e.  { A } ) )
21biimpi 188 . . . . 5  |-  ( x  e.  U. { A }  ->  E. q ( x  e.  q  /\  q  e.  { A } ) )
3 id 21 . . . . . . . . 9  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  (
x  e.  q  /\  q  e.  { A } ) )
4 simpl 445 . . . . . . . . 9  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  q )
53, 4syl 17 . . . . . . . 8  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  q )
6 id 21 . . . . . . . . . 10  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  (
x  e.  q  /\  q  e.  { A } ) )
7 simpr 449 . . . . . . . . . 10  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  q  e.  { A } )
86, 7syl 17 . . . . . . . . 9  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  q  e.  { A } )
9 elsni 3605 . . . . . . . . 9  |-  ( q  e.  { A }  ->  q  =  A )
108, 9syl 17 . . . . . . . 8  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  q  =  A )
11 eleq2 2317 . . . . . . . . 9  |-  ( q  =  A  ->  (
x  e.  q  <->  x  e.  A ) )
1211biimpac 474 . . . . . . . 8  |-  ( ( x  e.  q  /\  q  =  A )  ->  x  e.  A )
135, 10, 12syl2anc 645 . . . . . . 7  |-  ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A )
1413ax-gen 1536 . . . . . 6  |-  A. q
( ( x  e.  q  /\  q  e. 
{ A } )  ->  x  e.  A
)
15 19.23v 2022 . . . . . . 7  |-  ( A. q ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A
)  <->  ( E. q
( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A ) )
1615biimpi 188 . . . . . 6  |-  ( A. q ( ( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A
)  ->  ( E. q ( x  e.  q  /\  q  e. 
{ A } )  ->  x  e.  A
) )
1714, 16ax-mp 10 . . . . 5  |-  ( E. q ( x  e.  q  /\  q  e. 
{ A } )  ->  x  e.  A
)
18 pm3.35 573 . . . . 5  |-  ( ( E. q ( x  e.  q  /\  q  e.  { A } )  /\  ( E. q
( x  e.  q  /\  q  e.  { A } )  ->  x  e.  A ) )  ->  x  e.  A )
192, 17, 18sylancl 646 . . . 4  |-  ( x  e.  U. { A }  ->  x  e.  A
)
2019ax-gen 1536 . . 3  |-  A. x
( x  e.  U. { A }  ->  x  e.  A )
21 dfss2 3111 . . . 4  |-  ( U. { A }  C_  A  <->  A. x ( x  e. 
U. { A }  ->  x  e.  A ) )
2221biimpri 199 . . 3  |-  ( A. x ( x  e. 
U. { A }  ->  x  e.  A )  ->  U. { A }  C_  A )
2320, 22ax-mp 10 . 2  |-  U. { A }  C_  A
24 id 21 . . . . 5  |-  ( x  e.  A  ->  x  e.  A )
25 unisnALT.1 . . . . . 6  |-  A  e. 
_V
2625snid 3608 . . . . 5  |-  A  e. 
{ A }
27 elunii 3773 . . . . 5  |-  ( ( x  e.  A  /\  A  e.  { A } )  ->  x  e.  U. { A }
)
2824, 26, 27sylancl 646 . . . 4  |-  ( x  e.  A  ->  x  e.  U. { A }
)
2928ax-gen 1536 . . 3  |-  A. x
( x  e.  A  ->  x  e.  U. { A } )
30 dfss2 3111 . . . 4  |-  ( A 
C_  U. { A }  <->  A. x ( x  e.  A  ->  x  e.  U. { A } ) )
3130biimpri 199 . . 3  |-  ( A. x ( x  e.  A  ->  x  e.  U. { A } )  ->  A  C_  U. { A } )
3229, 31ax-mp 10 . 2  |-  A  C_  U. { A }
3323, 32eqssi 3137 1  |-  U. { A }  =  A
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   A.wal 1532   E.wex 1537    = wceq 1619    e. wcel 1621   _Vcvv 2740    C_ wss 3094   {csn 3581   U.cuni 3768
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-in 3101  df-ss 3108  df-sn 3587  df-uni 3769
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