Theorem opth 4271

Description: The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that C and D are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)

Hypotheses

Ref	Expression
opth1.1	|- A e. _V
opth1.2	|- B e. _V

Assertion

Ref	Expression
opth	|- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) )

Proof of Theorem opth

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opth1.1	. . . 4 |- A e. _V
2		opth1.2	. . . 4 |- B e. _V
3	1, 2	opth1 4270	. . 3 |- ( <. A , B >. = <. C , D >. -> A = C )
4	1, 2	opi1 4266	. . . . . . 7 |- { A } e. <. A , B >.
5		id 19	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> <. A , B >. = <. C , D >. )
6	4, 5	eleqtrid 2285	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> { A } e. <. C , D >. )
7		oprcl 3833	. . . . . 6 |- ( { A } e. <. C , D >. -> ( C e. _V /\ D e. _V ) )
8	6, 7	syl 14	. . . . 5 |- ( <. A , B >. = <. C , D >. -> ( C e. _V /\ D e. _V ) )
9	8	simprd 114	. . . 4 |- ( <. A , B >. = <. C , D >. -> D e. _V )
10	3	opeq1d 3815	. . . . . . . 8 |- ( <. A , B >. = <. C , D >. -> <. A , B >. = <. C , B >. )
11	10, 5	eqtr3d 2231	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> <. C , B >. = <. C , D >. )
12	8	simpld 112	. . . . . . . 8 |- ( <. A , B >. = <. C , D >. -> C e. _V )
13		dfopg 3807	. . . . . . . 8 |- ( ( C e. _V /\ B e. _V ) -> <. C , B >. = { { C } , { C , B } } )
14	12, 2, 13	sylancl 413	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> <. C , B >. = { { C } , { C , B } } )
15	11, 14	eqtr3d 2231	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> <. C , D >. = { { C } , { C , B } } )
16		dfopg 3807	. . . . . . 7 |- ( ( C e. _V /\ D e. _V ) -> <. C , D >. = { { C } , { C , D } } )
17	8, 16	syl 14	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> <. C , D >. = { { C } , { C , D } } )
18	15, 17	eqtr3d 2231	. . . . 5 |- ( <. A , B >. = <. C , D >. -> { { C } , { C , B } } = { { C } , { C , D } } )
19		prexg 4245	. . . . . . 7 |- ( ( C e. _V /\ B e. _V ) -> { C , B } e. _V )
20	12, 2, 19	sylancl 413	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> { C , B } e. _V )
21		prexg 4245	. . . . . . 7 |- ( ( C e. _V /\ D e. _V ) -> { C , D } e. _V )
22	8, 21	syl 14	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> { C , D } e. _V )
23		preqr2g 3798	. . . . . 6 |- ( ( { C , B } e. _V /\ { C , D } e. _V ) -> ( { { C } , { C , B } } = { { C } , { C , D } } -> { C , B } = { C , D } ) )
24	20, 22, 23	syl2anc 411	. . . . 5 |- ( <. A , B >. = <. C , D >. -> ( { { C } , { C , B } } = { { C } , { C , D } } -> { C , B } = { C , D } ) )
25	18, 24	mpd 13	. . . 4 |- ( <. A , B >. = <. C , D >. -> { C , B } = { C , D } )
26		preq2 3701	. . . . . . 7 |- ( x = D -> { C , x } = { C , D } )
27	26	eqeq2d 2208	. . . . . 6 |- ( x = D -> ( { C , B } = { C , x } <-> { C , B } = { C , D } ) )
28		eqeq2 2206	. . . . . 6 |- ( x = D -> ( B = x <-> B = D ) )
29	27, 28	imbi12d 234	. . . . 5 |- ( x = D -> ( ( { C , B } = { C , x } -> B = x ) <-> ( { C , B } = { C , D } -> B = D ) ) )
30		vex 2766	. . . . . 6 |- x e. _V
31	2, 30	preqr2 3800	. . . . 5 |- ( { C , B } = { C , x } -> B = x )
32	29, 31	vtoclg 2824	. . . 4 |- ( D e. _V -> ( { C , B } = { C , D } -> B = D ) )
33	9, 25, 32	sylc 62	. . 3 |- ( <. A , B >. = <. C , D >. -> B = D )
34	3, 33	jca 306	. 2 |- ( <. A , B >. = <. C , D >. -> ( A = C /\ B = D ) )
35		opeq12 3811	. 2 |- ( ( A = C /\ B = D ) -> <. A , B >. = <. C , D >. )
36	34, 35	impbii 126	1 |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   _Vcvv 2763   {csn 3623   {cpr 3624   <.cop 3626

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 4152  ax-pow 4208  ax-pr 4243

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 3608  df-sn 3629  df-pr 3630  df-op 3632

This theorem is referenced by:  opthg  4272  otth2  4275  copsexg  4278  copsex4g  4281  opcom  4284  moop2  4285  opelopabsbALT  4294  opelopabsb  4295  ralxpf  4813  rexxpf  4814  cnvcnvsn  5147  funopg  5293  funinsn  5308  brabvv  5972  xpdom2  6899  xpf1o  6914  djuf1olem  7128  enq0ref  7517  enq0tr  7518  mulnnnq0  7534  eqresr  7920  cnref1o  9742  fisumcom2  11620  fprodcom2fi  11808  qredeu  12290  fnpr2ob  13042