Theorem opi2 4938

Description: One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)

Hypotheses

Ref	Expression
opi1.1	|- A e. _V
opi1.2	|- B e. _V

Assertion

Ref	Expression
opi2	|- { A , B } e. <. A , B >.

Proof of Theorem opi2

Step	Hyp	Ref	Expression
1		prex 4909	. . 3 |- { A , B } e. _V
2	1	prid2 4298	. 2 |- { A , B } e. { { A } , { A , B } }
3		opi1.1	. . 3 |- A e. _V
4		opi1.2	. . 3 |- B e. _V
5	3, 4	dfop 4401	. 2 |- <. A , B >. = { { A } , { A , B } }
6	2, 5	eleqtrri 2700	1 |- { A , B } e. <. A , B >.

Syntax hints:   e. wcel 1990   _Vcvv 3200   {csn 4177   {cpr 4179   <.cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by:  opeluu  4939  uniopel  4978  elvvuni  5179