Theorem opi1 5470

Description: One of the two elements in an ordered pair. (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)

Hypotheses

Ref	Expression
opi1.1	⊢ 𝐴 ∈ V
opi1.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
opi1	⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩

Proof of Theorem opi1

Step	Hyp	Ref	Expression
1		snex 5433	. . 3 ⊢ {𝐴} ∈ V
2	1	prid1 4768	. 2 ⊢ {𝐴} ∈ {{𝐴}, {𝐴, 𝐵}}
3		opi1.1	. . 3 ⊢ 𝐴 ∈ V
4		opi1.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	dfop 4874	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
6	2, 5	eleqtrri 2824	1 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩

Syntax hints:   ∈ wcel 2098  Vcvv 3461  {csn 4630  {cpr 4632  ⟨cop 4636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637

This theorem is referenced by:  opth1  5477  opth  5478