Theorem opthg 4373

Description: Ordered pair theorem. C and D are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opthg	|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )

Proof of Theorem opthg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3899	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	eqeq1d 2247	. . 3 |- ( x = A -> ( <. x , y >. = <. C , D >. <-> <. A , y >. = <. C , D >. ) )
3		eqeq1 2245	. . . 4 |- ( x = A -> ( x = C <-> A = C ) )
4	3	anbi1d 469	. . 3 |- ( x = A -> ( ( x = C /\ y = D ) <-> ( A = C /\ y = D ) ) )
5	2, 4	bibi12d 235	. 2 |- ( x = A -> ( ( <. x , y >. = <. C , D >. <-> ( x = C /\ y = D ) ) <-> ( <. A , y >. = <. C , D >. <-> ( A = C /\ y = D ) ) ) )
6		opeq2 3900	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
7	6	eqeq1d 2247	. . 3 |- ( y = B -> ( <. A , y >. = <. C , D >. <-> <. A , B >. = <. C , D >. ) )
8		eqeq1 2245	. . . 4 |- ( y = B -> ( y = D <-> B = D ) )
9	8	anbi2d 468	. . 3 |- ( y = B -> ( ( A = C /\ y = D ) <-> ( A = C /\ B = D ) ) )
10	7, 9	bibi12d 235	. 2 |- ( y = B -> ( ( <. A , y >. = <. C , D >. <-> ( A = C /\ y = D ) ) <-> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) ) )
11		vex 2824	. . 3 |- x e. _V
12		vex 2824	. . 3 |- y e. _V
13	11, 12	opth 4372	. 2 |- ( <. x , y >. = <. C , D >. <-> ( x = C /\ y = D ) )
14	5, 10, 13	vtocl2g 2887	1 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   <.cop 3708

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714

This theorem is referenced by:  opthg2  4374  xpopth  6400  eqop  6401  inl11  7395  preqlu  7829  cauappcvgprlemladd  8015  elrealeu  8186  s111  11377  qnumdenbi  12948  crth  12980  imasaddfnlemg  13612  pellexlem3  16007