Theorem opth 4270

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 4269	. . 3 |- ( <. A , B >. = <. C , D >. -> A = C )
4	1, 2	opi1 4265	. . . . . . 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 3832	. . . . . 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 3814	. . . . . . . 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 3806	. . . . . . . 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 3806	. . . . . . 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 4244	. . . . . . 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 4244	. . . . . . 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 3797	. . . . . 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 3700	. . . . . . 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 3799	. . . . 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 3810	. 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 3622   {cpr 3623   <.cop 3625

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 4151  ax-pow 4207  ax-pr 4242

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 3607  df-sn 3628  df-pr 3629  df-op 3631

This theorem is referenced by:  opthg  4271  otth2  4274  copsexg  4277  copsex4g  4280  opcom  4283  moop2  4284  opelopabsbALT  4293  opelopabsb  4294  ralxpf  4812  rexxpf  4813  cnvcnvsn  5146  funopg  5292  funinsn  5307  brabvv  5968  xpdom2  6890  xpf1o  6905  djuf1olem  7119  enq0ref  7500  enq0tr  7501  mulnnnq0  7517  eqresr  7903  cnref1o  9725  fisumcom2  11603  fprodcom2fi  11791  qredeu  12265  fnpr2ob  12983