Theorem opi2 4865

Description: One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-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
opi2	⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩

Proof of Theorem opi2

Step	Hyp	Ref	Expression
1		prex 4836	. . 3 ⊢ {𝐴, 𝐵} ∈ V
2	1	prid2 4242	. 2 ⊢ {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}}
3		opi1.1	. . 3 ⊢ 𝐴 ∈ V
4		opi1.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	dfop 4339	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
6	2, 5	eleqtrri 2687	1 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩

Syntax hints:   ∈ wcel 1977  Vcvv 3173  {csn 4125  {cpr 4127  ⟨cop 4131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132

This theorem is referenced by:  opeluu  4866  uniopel  4903  elvvuni  5102