Theorem opi2 4248

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 4226	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
4	1, 2, 3	mp2an 426	. . 3 ⊢ {𝐴, 𝐵} ∈ V
5	4	prid2 3714	. 2 ⊢ {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}}
6	1, 2	dfop 3792	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
7	5, 6	eleqtrri 2265	1 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩

Syntax hints:   ∈ wcel 2160  Vcvv 2752  {csn 3607  {cpr 3608  ⟨cop 3610

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616

This theorem is referenced by:  uniopel  4271  opeluu  4465  elvvuni  4705