Theorem opth2 2800

Description: Equality of the second members of equal ordered pairs. Because of our particular ordered pair definition, equality holds whether or not the first members are sets.

Hypotheses

Ref	Expression
opth2.1	|- B e. V
opth2.2	|- D e. V

Assertion

Ref	Expression
opth2	|- (<.A, B>. = <.C, D>. -> B = D)

Proof of Theorem opth2

Step	Hyp	Ref	Expression
1		opeq1 2487	. . . . 5 |- (x = A -> <.x, B>. = <.A, B>.)
2	1	eqeq1d 1483	. . . 4 |- (x = A -> (<.x, B>. = <.C, D>. <-> <.A, B>. = <.C, D>.))
3	2	imbi1d 613	. . 3 |- (x = A -> ((<.x, B>. = <.C, D>. -> B = D) <-> (<.A, B>. = <.C, D>. -> B = D)))
4		visset 1813	. . . . 5 |- x e. V
5		opth2.1	. . . . 5 |- B e. V
6		opth2.2	. . . . 5 |- D e. V
7	4, 5, 6	opth 2787	. . . 4 |- (<.x, B>. = <.C, D>. <-> (x = C /\ B = D))
8	7	pm3.27bi 326	. . 3 |- (<.x, B>. = <.C, D>. -> B = D)
9	3, 8	vtoclg 1847	. 2 |- (A e. V -> (<.A, B>. = <.C, D>. -> B = D))
10		nelneq2 1562	. . . . 5 |- (((/) e. <.A, B>. /\ -. (/) e. <.C, D>.) -> -. <.A, B>. = <.C, D>.)
11		opprc1b 2796	. . . . 5 |- (-. A e. V <-> (/) e. <.A, B>.)
12		opprc1b 2796	. . . . . . 7 |- (-. C e. V <-> (/) e. <.C, D>.)
13	12	con1bii 220	. . . . . 6 |- (-. (/) e. <.C, D>. <-> C e. V)
14	13	bicomi 172	. . . . 5 |- (C e. V <-> -. (/) e. <.C, D>.)
15	10, 11, 14	syl2anb 455	. . . 4 |- ((-. A e. V /\ C e. V) -> -. <.A, B>. = <.C, D>.)
16	15	pm2.21d 78	. . 3 |- ((-. A e. V /\ C e. V) -> (<.A, B>. = <.C, D>. -> B = D))
17		opprc1 2498	. . . . 5 |- (-. A e. V -> <.A, B>. = {(/), {B}})
18		opprc1 2498	. . . . 5 |- (-. C e. V -> <.C, D>. = {(/), {D}})
19	17, 18	eqeqan12d 1490	. . . 4 |- ((-. A e. V /\ -. C e. V) -> (<.A, B>. = <.C, D>. <-> {(/), {B}} = {(/), {D}}))
20		snex 2750	. . . . . 6 |- {B} e. V
21		snex 2750	. . . . . 6 |- {D} e. V
22	20, 21	preqr2 2482	. . . . 5 |- ({(/), {B}} = {(/), {D}} -> {B} = {D})
23	5	sneqr 2477	. . . . 5 |- ({B} = {D} -> B = D)
24	22, 23	syl 10	. . . 4 |- ({(/), {B}} = {(/), {D}} -> B = D)
25	19, 24	syl6bi 214	. . 3 |- ((-. A e. V /\ -. C e. V) -> (<.A, B>. = <.C, D>. -> B = D))
26	16, 25	pm2.61dan 477	. 2 |- (-. A e. V -> (<.A, B>. = <.C, D>. -> B = D))
27	9, 26	pm2.61i 126	1 |- (<.A, B>. = <.C, D>. -> B = D)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  (/)c0 2280  {csn 2409  {cpr 2410  <.cop 2411

This theorem is referenced by:  moop2 2801  funsn 3543

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416

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