Theorem opi1 4262

Description: One of the two elements in 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
opi1	⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩

Proof of Theorem opi1

Step	Hyp	Ref	Expression
1		opi1.1	. . . 4 ⊢ 𝐴 ∈ V
2	1	snex 4215	. . 3 ⊢ {𝐴} ∈ V
3	2	prid1 3725	. 2 ⊢ {𝐴} ∈ {{𝐴}, {𝐴, 𝐵}}
4		opi1.2	. . 3 ⊢ 𝐵 ∈ V
5	1, 4	dfop 3804	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
6	3, 5	eleqtrri 2269	1 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩

Syntax hints:   ∈ wcel 2164  Vcvv 2760  {csn 3619  {cpr 3620  ⟨cop 3622

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 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628

This theorem is referenced by:  opth1  4266  opth  4267