Theorem opi1 5417

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 5382	. . 3 ⊢ {𝐴} ∈ V
2	1	prid1 4720	. 2 ⊢ {𝐴} ∈ {{𝐴}, {𝐴, 𝐵}}
3		opi1.1	. . 3 ⊢ 𝐴 ∈ V
4		opi1.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	dfop 4829	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
6	2, 5	eleqtrri 2836	1 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩

Syntax hints:   ∈ wcel 2114  Vcvv 3441  {csn 4581  {cpr 4583  ⟨cop 4587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588

This theorem is referenced by:  opth1  5424  opth  5425