Theorem opi2 4211

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 4189	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
4	1, 2, 3	mp2an 423	. . 3 ⊢ {𝐴, 𝐵} ∈ V
5	4	prid2 3683	. 2 ⊢ {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}}
6	1, 2	dfop 3757	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
7	5, 6	eleqtrri 2242	1 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩

Syntax hints:   ∈ wcel 2136  Vcvv 2726  {csn 3576  {cpr 3577  ⟨cop 3579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585

This theorem is referenced by:  uniopel  4234  opeluu  4428  elvvuni  4668