Theorem opthg 4155

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 3700	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	eqeq1d 2146	. . 3 |- ( x = A -> ( <. x , y >. = <. C , D >. <-> <. A , y >. = <. C , D >. ) )
3		eqeq1 2144	. . . 4 |- ( x = A -> ( x = C <-> A = C ) )
4	3	anbi1d 460	. . 3 |- ( x = A -> ( ( x = C /\ y = D ) <-> ( A = C /\ y = D ) ) )
5	2, 4	bibi12d 234	. 2 |- ( x = A -> ( ( <. x , y >. = <. C , D >. <-> ( x = C /\ y = D ) ) <-> ( <. A , y >. = <. C , D >. <-> ( A = C /\ y = D ) ) ) )
6		opeq2 3701	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
7	6	eqeq1d 2146	. . 3 |- ( y = B -> ( <. A , y >. = <. C , D >. <-> <. A , B >. = <. C , D >. ) )
8		eqeq1 2144	. . . 4 |- ( y = B -> ( y = D <-> B = D ) )
9	8	anbi2d 459	. . 3 |- ( y = B -> ( ( A = C /\ y = D ) <-> ( A = C /\ B = D ) ) )
10	7, 9	bibi12d 234	. 2 |- ( y = B -> ( ( <. A , y >. = <. C , D >. <-> ( A = C /\ y = D ) ) <-> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) ) )
11		vex 2684	. . 3 |- x e. _V
12		vex 2684	. . 3 |- y e. _V
13	11, 12	opth 4154	. 2 |- ( <. x , y >. = <. C , D >. <-> ( x = C /\ y = D ) )
14	5, 10, 13	vtocl2g 2745	1 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   <.cop 3525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531

This theorem is referenced by:  opthg2  4156  xpopth  6067  eqop  6068  inl11  6943  preqlu  7273  cauappcvgprlemladd  7459  elrealeu  7630  qnumdenbi  11859  crth  11889