Theorem opth 4154

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 4153	. . 3 |- ( <. A , B >. = <. C , D >. -> A = C )
4	1, 2	opi1 4149	. . . . . . 7 |- { A } e. <. A , B >.
5		id 19	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> <. A , B >. = <. C , D >. )
6	4, 5	eleqtrid 2226	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> { A } e. <. C , D >. )
7		oprcl 3724	. . . . . 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 3706	. . . . . . . 8 |- ( <. A , B >. = <. C , D >. -> <. A , B >. = <. C , B >. )
11	10, 5	eqtr3d 2172	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> <. C , B >. = <. C , D >. )
12	8	simpld 111	. . . . . . . 8 |- ( <. A , B >. = <. C , D >. -> C e. _V )
13		dfopg 3698	. . . . . . . 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 2172	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> <. C , D >. = { { C } , { C , B } } )
16		dfopg 3698	. . . . . . 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 2172	. . . . 5 |- ( <. A , B >. = <. C , D >. -> { { C } , { C , B } } = { { C } , { C , D } } )
19		prexg 4128	. . . . . . 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 4128	. . . . . . 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 3689	. . . . . 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 3596	. . . . . . 7 |- ( x = D -> { C , x } = { C , D } )
27	26	eqeq2d 2149	. . . . . 6 |- ( x = D -> ( { C , B } = { C , x } <-> { C , B } = { C , D } ) )
28		eqeq2 2147	. . . . . 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 2684	. . . . . 6 |- x e. _V
31	2, 30	preqr2 3691	. . . . 5 |- ( { C , B } = { C , x } -> B = x )
32	29, 31	vtoclg 2741	. . . 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 3702	. 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 2681   {csn 3522   {cpr 3523   <.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:  opthg  4155  otth2  4158  copsexg  4161  copsex4g  4164  opcom  4167  moop2  4168  opelopabsbALT  4176  opelopabsb  4177  ralxpf  4680  rexxpf  4681  cnvcnvsn  5010  funopg  5152  funinsn  5167  brabvv  5810  xpdom2  6718  xpf1o  6731  djuf1olem  6931  enq0ref  7234  enq0tr  7235  mulnnnq0  7251  eqresr  7637  cnref1o  9433  fisumcom2  11200  qredeu  11767