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Theorem opnzi 4000
Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 3999). (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 3999 . . 3  |-  ( E. x  x  e.  <. A ,  B >.  <->  ( A  e.  _V  /\  B  e. 
_V ) )
41, 2, 3mpbir2an 860 . 2  |-  E. x  x  e.  <. A ,  B >.
5 n0r 3262 . 2  |-  ( E. x  x  e.  <. A ,  B >.  ->  <. A ,  B >.  =/=  (/) )
64, 5ax-mp 7 1  |-  <. A ,  B >.  =/=  (/)
Colors of variables: wff set class
Syntax hints:   E.wex 1397    e. wcel 1409    =/= wne 2220   _Vcvv 2574   (/)c0 3252   <.cop 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412
This theorem is referenced by:  0nelxp  4400  0neqopab  5578
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