Theorem opi2 4349

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 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		opi1.1	. . . 4 |- A e. _V
2		opi1.2	. . . 4 |- B e. _V
3		prexg 4325	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
4	1, 2, 3	mp2an 426	. . 3 |- { A , B } e. _V
5	4	prid2 3798	. 2 |- { A , B } e. { { A } , { A , B } }
6	1, 2	dfop 3882	. 2 |- <. A , B >. = { { A } , { A , B } }
7	5, 6	eleqtrri 2308	1 |- { A , B } e. <. A , B >.

Syntax hints:   e. wcel 2203   _Vcvv 2813   {csn 3689   {cpr 3690   <.cop 3692

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 2206  ax-ext 2214  ax-sep 4228  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698

This theorem is referenced by:  uniopel  4373  opeluu  4571  elvvuni  4814