Theorem opi1 4330

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 e. _V
opi1.2	|- B e. _V

Assertion

Ref	Expression
opi1	|- { A } e. <. A , B >.

Proof of Theorem opi1

Step	Hyp	Ref	Expression
1		opi1.1	. . . 4 |- A e. _V
2	1	snex 4281	. . 3 |- { A } e. _V
3	2	prid1 3781	. 2 |- { A } e. { { A } , { A , B } }
4		opi1.2	. . 3 |- B e. _V
5	1, 4	dfop 3866	. 2 |- <. A , B >. = { { A } , { A , B } }
6	3, 5	eleqtrri 2307	1 |- { A } e. <. A , B >.

Syntax hints:   e. wcel 2202   _Vcvv 2803   {csn 3673   {cpr 3674   <.cop 3676

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682

This theorem is referenced by:  opth1  4334  opth  4335