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Theorem opnzi 3963
Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 3962). (Contributed by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opth1.1 A V
opth1.2 B 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 V
2 opth1.2 . . 3 B V
3 opm 3962 . . 3 (x x A, B⟩ ↔ (A V B V))
41, 2, 3mpbir2an 848 . 2 x x A, B
5 n0r 3228 . 2 (x x A, B⟩ → ⟨A, B⟩ ≠ ∅)
64, 5ax-mp 7 1 A, B⟩ ≠ ∅
Colors of variables: wff set class
Syntax hints:  wex 1378   wcel 1390  wne 2201  Vcvv 2551  c0 3218  cop 3370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376
This theorem is referenced by:  0nelxp  4315  0neqopab  5492
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