Theorem opth 4159

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 4158	. . 3 |- ( <. A , B >. = <. C , D >. -> A = C )
4	1, 2	opi1 4154	. . . . . . 7 |- { A } e. <. A , B >.
5		id 19	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> <. A , B >. = <. C , D >. )
6	4, 5	eleqtrid 2228	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> { A } e. <. C , D >. )
7		oprcl 3729	. . . . . 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 113	. . . 4 |- ( <. A , B >. = <. C , D >. -> D e. _V )
10	3	opeq1d 3711	. . . . . . . 8 |- ( <. A , B >. = <. C , D >. -> <. A , B >. = <. C , B >. )
11	10, 5	eqtr3d 2174	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> <. C , B >. = <. C , D >. )
12	8	simpld 111	. . . . . . . 8 |- ( <. A , B >. = <. C , D >. -> C e. _V )
13		dfopg 3703	. . . . . . . 8 |- ( ( C e. _V /\ B e. _V ) -> <. C , B >. = { { C } , { C , B } } )
14	12, 2, 13	sylancl 409	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> <. C , B >. = { { C } , { C , B } } )
15	11, 14	eqtr3d 2174	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> <. C , D >. = { { C } , { C , B } } )
16		dfopg 3703	. . . . . . 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 2174	. . . . 5 |- ( <. A , B >. = <. C , D >. -> { { C } , { C , B } } = { { C } , { C , D } } )
19		prexg 4133	. . . . . . 7 |- ( ( C e. _V /\ B e. _V ) -> { C , B } e. _V )
20	12, 2, 19	sylancl 409	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> { C , B } e. _V )
21		prexg 4133	. . . . . . 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 3694	. . . . . 6 |- ( ( { C , B } e. _V /\ { C , D } e. _V ) -> ( { { C } , { C , B } } = { { C } , { C , D } } -> { C , B } = { C , D } ) )
24	20, 22, 23	syl2anc 408	. . . . 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 3601	. . . . . . 7 |- ( x = D -> { C , x } = { C , D } )
27	26	eqeq2d 2151	. . . . . 6 |- ( x = D -> ( { C , B } = { C , x } <-> { C , B } = { C , D } ) )
28		eqeq2 2149	. . . . . 6 |- ( x = D -> ( B = x <-> B = D ) )
29	27, 28	imbi12d 233	. . . . 5 |- ( x = D -> ( ( { C , B } = { C , x } -> B = x ) <-> ( { C , B } = { C , D } -> B = D ) ) )
30		vex 2689	. . . . . 6 |- x e. _V
31	2, 30	preqr2 3696	. . . . 5 |- ( { C , B } = { C , x } -> B = x )
32	29, 31	vtoclg 2746	. . . 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 304	. 2 |- ( <. A , B >. = <. C , D >. -> ( A = C /\ B = D ) )
35		opeq12 3707	. 2 |- ( ( A = C /\ B = D ) -> <. A , B >. = <. C , D >. )
36	34, 35	impbii 125	1 |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   _Vcvv 2686   {csn 3527   {cpr 3528   <.cop 3530

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536

This theorem is referenced by:  opthg  4160  otth2  4163  copsexg  4166  copsex4g  4169  opcom  4172  moop2  4173  opelopabsbALT  4181  opelopabsb  4182  ralxpf  4685  rexxpf  4686  cnvcnvsn  5015  funopg  5157  funinsn  5172  brabvv  5817  xpdom2  6725  xpf1o  6738  djuf1olem  6938  enq0ref  7241  enq0tr  7242  mulnnnq0  7258  eqresr  7644  cnref1o  9440  fisumcom2  11207  qredeu  11778