Theorem opi2 4290

Description: One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)

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		opi1.1	. . . 4 ⊢ 𝐴 ∈ V
2		opi1.2	. . . 4 ⊢ 𝐵 ∈ V
3		prexg 4266	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
4	1, 2, 3	mp2an 426	. . 3 ⊢ {𝐴, 𝐵} ∈ V
5	4	prid2 3745	. 2 ⊢ {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}}
6	1, 2	dfop 3827	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
7	5, 6	eleqtrri 2282	1 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩

Syntax hints:   ∈ wcel 2177  Vcvv 2773  {csn 3638  {cpr 3639  ⟨cop 3641

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647

This theorem is referenced by:  uniopel  4314  opeluu  4510  elvvuni  4752