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Theorem prmg 3713
Description: A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
Assertion
Ref Expression
prmg  |-  ( A  e.  V  ->  E. x  x  e.  { A ,  B } )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem prmg
StepHypRef Expression
1 snmg 3710 . 2  |-  ( A  e.  V  ->  E. x  x  e.  { A } )
2 orc 712 . . . 4  |-  ( x  =  A  ->  (
x  =  A  \/  x  =  B )
)
3 velsn 3609 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
4 vex 2740 . . . . 5  |-  x  e. 
_V
54elpr 3613 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
62, 3, 53imtr4i 201 . . 3  |-  ( x  e.  { A }  ->  x  e.  { A ,  B } )
76eximi 1600 . 2  |-  ( E. x  x  e.  { A }  ->  E. x  x  e.  { A ,  B } )
81, 7syl 14 1  |-  ( A  e.  V  ->  E. x  x  e.  { A ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 708    = wceq 1353   E.wex 1492    e. wcel 2148   {csn 3592   {cpr 3593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599
This theorem is referenced by:  prm  3715  opm  4234  onintexmid  4572  subrgin  13325
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