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Theorem opnzi 3972
 Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 3971). (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 3971 . . 3 (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
41, 2, 3mpbir2an 849 . 2 𝑥 𝑥 ∈ ⟨𝐴, 𝐵
5 n0r 3234 . 2 (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ ≠ ∅)
64, 5ax-mp 7 1 𝐴, 𝐵⟩ ≠ ∅
 Colors of variables: wff set class Syntax hints:  ∃wex 1381   ∈ wcel 1393   ≠ wne 2204  Vcvv 2557  ∅c0 3224  ⟨cop 3378 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384 This theorem is referenced by:  0nelxp  4372  0neqopab  5550
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