Theorem opi1 5455

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 5415	. . 3 ⊢ {𝐴} ∈ V
2	1	prid1 4733	. 2 ⊢ {𝐴} ∈ {{𝐴}, {𝐴, 𝐵}}
3		opi1.1	. . 3 ⊢ 𝐴 ∈ V
4		opi1.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	dfop 4842	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
6	2, 5	eleqtrri 2865	1 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩

This proof depends on syntax axioms:   ∈ wcel 2146  Vcvv 3458  {csn 4594  {cpr 4596  ⟨cop 4600

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601

This theorem is used by:  opth1  5462  opth  5463