Theorem opi1 4276

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 4229	. . 3 ⊢ {𝐴} ∈ V
3	2	prid1 3739	. 2 ⊢ {𝐴} ∈ {{𝐴}, {𝐴, 𝐵}}
4		opi1.2	. . 3 ⊢ 𝐵 ∈ V
5	1, 4	dfop 3818	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
6	3, 5	eleqtrri 2281	1 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩

Syntax hints:   ∈ wcel 2176  Vcvv 2772  {csn 3633  {cpr 3634  ⟨cop 3636

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642

This theorem is referenced by:  opth1  4280  opth  4281