Theorem opthg 4283

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 3819	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	eqeq1d 2214	. . 3 |- ( x = A -> ( <. x , y >. = <. C , D >. <-> <. A , y >. = <. C , D >. ) )
3		eqeq1 2212	. . . 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 3820	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
7	6	eqeq1d 2214	. . 3 |- ( y = B -> ( <. A , y >. = <. C , D >. <-> <. A , B >. = <. C , D >. ) )
8		eqeq1 2212	. . . 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 2775	. . 3 |- x e. _V
12		vex 2775	. . 3 |- y e. _V
13	11, 12	opth 4282	. 2 |- ( <. x , y >. = <. C , D >. <-> ( x = C /\ y = D ) )
14	5, 10, 13	vtocl2g 2837	1 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   <.cop 3636

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642

This theorem is referenced by:  opthg2  4284  xpopth  6264  eqop  6265  inl11  7169  preqlu  7587  cauappcvgprlemladd  7773  elrealeu  7944  s111  11088  qnumdenbi  12547  crth  12579  imasaddfnlemg  13179