Theorem opth 4379

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 4378	. . 3 |- ( <. A , B >. = <. C , D >. -> A = C )
4	1, 2	opi1 4374	. . . . . . 7 |- { A } e. <. A , B >.
5		id 19	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> <. A , B >. = <. C , D >. )
6	4, 5	eleqtrid 2327	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> { A } e. <. C , D >. )
7		oprcl 3928	. . . . . 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 3910	. . . . . . . 8 |- ( <. A , B >. = <. C , D >. -> <. A , B >. = <. C , B >. )
11	10, 5	eqtr3d 2273	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> <. C , B >. = <. C , D >. )
12	8	simpld 112	. . . . . . . 8 |- ( <. A , B >. = <. C , D >. -> C e. _V )
13		dfopg 3902	. . . . . . . 8 |- ( ( C e. _V /\ B e. _V ) -> <. C , B >. = { { C } , { C , B } } )
14	12, 2, 13	sylancl 417	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> <. C , B >. = { { C } , { C , B } } )
15	11, 14	eqtr3d 2273	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> <. C , D >. = { { C } , { C , B } } )
16		dfopg 3902	. . . . . . 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 2273	. . . . 5 |- ( <. A , B >. = <. C , D >. -> { { C } , { C , B } } = { { C } , { C , D } } )
19		prexg 4351	. . . . . . 7 |- ( ( C e. _V /\ B e. _V ) -> { C , B } e. _V )
20	12, 2, 19	sylancl 417	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> { C , B } e. _V )
21		prexg 4351	. . . . . . 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 3892	. . . . . 6 |- ( ( { C , B } e. _V /\ { C , D } e. _V ) -> ( { { C } , { C , B } } = { { C } , { C , D } } -> { C , B } = { C , D } ) )
24	20, 22, 23	syl2anc 415	. . . . 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 3789	. . . . . . 7 |- ( x = D -> { C , x } = { C , D } )
27	26	eqeq2d 2250	. . . . . 6 |- ( x = D -> ( { C , B } = { C , x } <-> { C , B } = { C , D } ) )
28		eqeq2 2248	. . . . . 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 2824	. . . . . 6 |- x e. _V
31	2, 30	preqr2 3894	. . . . 5 |- ( { C , B } = { C , x } -> B = x )
32	29, 31	vtoclg 2883	. . . 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 3906	. 2 |- ( ( A = C /\ B = D ) -> <. A , B >. = <. C , D >. )
36	34, 35	impbii 126	1 |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3709   {cpr 3710   <.cop 3712

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718

This theorem is used by:  opthg  4380  otth2  4383  copsexg  4386  copsex4g  4389  opcom  4393  moop2  4394  opelopabsbALT  4403  opelopabsb  4404  ralxpf  4928  rexxpf  4929  cnvcnvsn  5267  funopg  5414  funinsn  5433  brabvv  6137  xpdom2  7133  xpf1o  7148  djuf1olem  7399  enq0ref  7806  enq0tr  7807  mulnnnq0  7823  eqresr  8209  cnref1o  10068  fisumcom2  12230  fprodcom2fi  12418  qredeu  12900  fnpr2ob  13720