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Theorem opnzi 4231
Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4230). (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 4230 . . 3 (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
41, 2, 3mpbir2an 942 . 2 𝑥 𝑥 ∈ ⟨𝐴, 𝐵
5 n0r 3436 . 2 (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ ≠ ∅)
64, 5ax-mp 5 1 𝐴, 𝐵⟩ ≠ ∅
Colors of variables: wff set class
Syntax hints:  wex 1492  wcel 2148  wne 2347  Vcvv 2737  c0 3422  cop 3594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600
This theorem is referenced by:  0nelxp  4650  0neqopab  5913
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