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Theorem opnzi 4210
Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4209). (Contributed by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opth1.1  |-  A  e. 
_V
opth1.2  |-  B  e. 
_V
Assertion
Ref Expression
opnzi  |-  <. A ,  B >.  =/=  (/)

Proof of Theorem opnzi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 opth1.1 . . 3  |-  A  e. 
_V
2 opth1.2 . . 3  |-  B  e. 
_V
3 opm 4209 . . 3  |-  ( E. x  x  e.  <. A ,  B >.  <->  ( A  e.  _V  /\  B  e. 
_V ) )
41, 2, 3mpbir2an 931 . 2  |-  E. x  x  e.  <. A ,  B >.
5 n0r 3420 . 2  |-  ( E. x  x  e.  <. A ,  B >.  ->  <. A ,  B >.  =/=  (/) )
64, 5ax-mp 5 1  |-  <. A ,  B >.  =/=  (/)
Colors of variables: wff set class
Syntax hints:   E.wex 1479    e. wcel 2135    =/= wne 2334   _Vcvv 2724   (/)c0 3407   <.cop 3576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-v 2726  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3408  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582
This theorem is referenced by:  0nelxp  4629  0neqopab  5881
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