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Theorem opnzi 4034
Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4033). (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 4033 . . 3 (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
41, 2, 3mpbir2an 886 . 2 𝑥 𝑥 ∈ ⟨𝐴, 𝐵
5 n0r 3285 . 2 (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ ≠ ∅)
64, 5ax-mp 7 1 𝐴, 𝐵⟩ ≠ ∅
Colors of variables: wff set class
Syntax hints:  wex 1424  wcel 1436  wne 2251  Vcvv 2615  c0 3275  cop 3433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3930  ax-pow 3982
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3416  df-sn 3436  df-pr 3437  df-op 3439
This theorem is referenced by:  0nelxp  4436  0neqopab  5644
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