Theorem opi2 4319

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 4295	. . . 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 3773	. 2 |- { A , B } e. { { A } , { A , B } }
6	1, 2	dfop 3856	. 2 |- <. A , B >. = { { A } , { A , B } }
7	5, 6	eleqtrri 2305	1 |- { A , B } e. <. A , B >.

Syntax hints:   e. wcel 2200   _Vcvv 2799   {csn 3666   {cpr 3667   <.cop 3669

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675

This theorem is referenced by:  uniopel  4343  opeluu  4541  elvvuni  4783