Theorem elopg 4934

Description: Characterization of the elements of an ordered pair. Closed form of elop 4935. (Contributed by BJ, 22-Jun-2019.) (Avoid depending on this detail.)

Assertion

Ref	Expression
elopg	|- ( ( A e. V /\ B e. W ) -> ( C e. <. A , B >. <-> ( C = { A } \/ C = { A , B } ) ) )

Proof of Theorem elopg

Step	Hyp	Ref	Expression
1		dfopg 4400	. 2 |- ( ( A e. V /\ B e. W ) -> <. A , B >. = { { A } , { A , B } } )
2		eleq2 2690	. . 3 |- ( <. A , B >. = { { A } , { A , B } } -> ( C e. <. A , B >. <-> C e. { { A } , { A , B } } ) )
3		snex 4908	. . . 4 |- { A } e. _V
4		prex 4909	. . . 4 |- { A , B } e. _V
5	3, 4	elpr2 4199	. . 3 |- ( C e. { { A } , { A , B } } <-> ( C = { A } \/ C = { A , B } ) )
6	2, 5	syl6bb 276	. 2 |- ( <. A , B >. = { { A } , { A , B } } -> ( C e. <. A , B >. <-> ( C = { A } \/ C = { A , B } ) ) )
7	1, 6	syl 17	1 |- ( ( A e. V /\ B e. W ) -> ( C e. <. A , B >. <-> ( C = { A } \/ C = { A , B } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {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:  elop  4935  bj-inftyexpidisj  33097