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Theorem opnzi 3999
Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 3998). (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 3998 . . 3 (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
41, 2, 3mpbir2an 860 . 2 𝑥 𝑥 ∈ ⟨𝐴, 𝐵
5 n0r 3261 . 2 (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ ≠ ∅)
64, 5ax-mp 7 1 𝐴, 𝐵⟩ ≠ ∅
Colors of variables: wff set class
Syntax hints:  wex 1397  wcel 1409  wne 2220  Vcvv 2574  c0 3251  cop 3405
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 3902  ax-pow 3954
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 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411
This theorem is referenced by:  0nelxp  4399  0neqopab  5577
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