Theorem opnzi 3990

Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 3989). (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.

Step	Hyp	Ref	Expression
1		opth1.1	. . 3 ⊢ 𝐴 ∈ V
2		opth1.2	. . 3 ⊢ 𝐵 ∈ V
3		opm 3989	. . 3 ⊢ (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	1, 2, 3	mpbir2an 883	. 2 ⊢ ∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩
5		n0r 3261	. 2 ⊢ (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ ≠ ∅)
6	4, 5	ax-mp 7	1 ⊢ ⟨𝐴, 𝐵⟩ ≠ ∅

Syntax hints:  ∃wex 1421   ∈ wcel 1433   ≠ wne 2245  Vcvv 2601  ∅c0 3251  ⟨cop 3401

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407

This theorem is referenced by:  0nelxp  4390  0neqopab  5570