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Theorem opnzi 4247
Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4246). (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 4246 . . 3 (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
41, 2, 3mpbir2an 943 . 2 𝑥 𝑥 ∈ ⟨𝐴, 𝐵
5 n0r 3448 . 2 (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ ≠ ∅)
64, 5ax-mp 5 1 𝐴, 𝐵⟩ ≠ ∅
Colors of variables: wff set class
Syntax hints:  wex 1502  wcel 2158  wne 2357  Vcvv 2749  c0 3434  cop 3607
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613
This theorem is referenced by:  0nelxp  4666  0neqopab  5933
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