Theorem opnzi 4351

Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4350). (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 4350	. . 3 ⊢ (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	1, 2, 3	mpbir2an 951	. 2 ⊢ ∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩
5		n0r 3522	. 2 ⊢ (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ ≠ ∅)
6	4, 5	ax-mp 5	1 ⊢ ⟨𝐴, 𝐵⟩ ≠ ∅

Syntax hints:  ∃wex 1541   ∈ wcel 2203   ≠ wne 2412  Vcvv 2813  ∅c0 3508  ⟨cop 3692

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698

This theorem is referenced by:  0nelxp  4777  0neqopab  6098