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Theorem prmg 3544
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 3541 . 2  |-  ( A  e.  V  ->  E. x  x  e.  { A } )
2 orc 666 . . . 4  |-  ( x  =  A  ->  (
x  =  A  \/  x  =  B )
)
3 velsn 3448 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
4 vex 2618 . . . . 5  |-  x  e. 
_V
54elpr 3452 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
62, 3, 53imtr4i 199 . . 3  |-  ( x  e.  { A }  ->  x  e.  { A ,  B } )
76eximi 1534 . 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 1287   E.wex 1424    e. wcel 1436   {csn 3431   {cpr 3432
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-sn 3437  df-pr 3438
This theorem is referenced by:  prm  3546  opm  4034  onintexmid  4360
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