Theorem opnzi 4278

Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4277). (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 4277	. . 3 ⊢ (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	1, 2, 3	mpbir2an 944	. 2 ⊢ ∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩
5		n0r 3473	. 2 ⊢ (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ ≠ ∅)
6	4, 5	ax-mp 5	1 ⊢ ⟨𝐴, 𝐵⟩ ≠ ∅

Syntax hints:  ∃wex 1514   ∈ wcel 2175   ≠ wne 2375  Vcvv 2771  ∅c0 3459  ⟨cop 3635

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641

This theorem is referenced by:  0nelxp  4702  0neqopab  5989