Theorem opi2 4331

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 4307	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
4	1, 2, 3	mp2an 426	. . 3 ⊢ {𝐴, 𝐵} ∈ V
5	4	prid2 3782	. 2 ⊢ {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}}
6	1, 2	dfop 3866	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
7	5, 6	eleqtrri 2307	1 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩

Syntax hints:   ∈ wcel 2202  Vcvv 2803  {csn 3673  {cpr 3674  ⟨cop 3676

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682

This theorem is referenced by:  uniopel  4355  opeluu  4553  elvvuni  4796