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Theorem prmg 3529
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 3526 . 2  |-  ( A  e.  V  ->  E. x  x  e.  { A } )
2 orc 666 . . . 4  |-  ( x  =  A  ->  (
x  =  A  \/  x  =  B )
)
3 velsn 3433 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
4 vex 2613 . . . . 5  |-  x  e. 
_V
54elpr 3437 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
62, 3, 53imtr4i 199 . . 3  |-  ( x  e.  { A }  ->  x  e.  { A ,  B } )
76eximi 1532 . 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 662    = wceq 1285   E.wex 1422    e. wcel 1434   {csn 3416   {cpr 3417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423
This theorem is referenced by:  prm  3531  opm  4017  onintexmid  4343
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