Theorem opthg 4271

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 3808	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	eqeq1d 2205	. . 3 |- ( x = A -> ( <. x , y >. = <. C , D >. <-> <. A , y >. = <. C , D >. ) )
3		eqeq1 2203	. . . 4 |- ( x = A -> ( x = C <-> A = C ) )
4	3	anbi1d 465	. . 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 3809	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
7	6	eqeq1d 2205	. . 3 |- ( y = B -> ( <. A , y >. = <. C , D >. <-> <. A , B >. = <. C , D >. ) )
8		eqeq1 2203	. . . 4 |- ( y = B -> ( y = D <-> B = D ) )
9	8	anbi2d 464	. . 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 2766	. . 3 |- x e. _V
12		vex 2766	. . 3 |- y e. _V
13	11, 12	opth 4270	. 2 |- ( <. x , y >. = <. C , D >. <-> ( x = C /\ y = D ) )
14	5, 10, 13	vtocl2g 2828	1 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   <.cop 3625

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631

This theorem is referenced by:  opthg2  4272  xpopth  6234  eqop  6235  inl11  7131  preqlu  7539  cauappcvgprlemladd  7725  elrealeu  7896  qnumdenbi  12360  crth  12392  imasaddfnlemg  12957