Theorem opth2 2765

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 2456	. . . . 5 |- (x = A -> <.x, B>. = <.A, B>.)
2	1	eqeq1d 1459	. . . 4 |- (x = A -> (<.x, B>. = <.C, D>. <-> <.A, B>. = <.C, D>.))
3	2	imbi1d 611	. . 3 |- (x = A -> ((<.x, B>. = <.C, D>. -> B = D) <-> (<.A, B>. = <.C, D>. -> B = D)))
4		visset 1788	. . . . 5 |- x e. V
5		opth2.1	. . . . 5 |- B e. V
6		opth2.2	. . . . 5 |- D e. V
7	4, 5, 6	opth 2754	. . . 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 1822	. 2 |- (A e. V -> (<.A, B>. = <.C, D>. -> B = D))
10		nelneq2 1538	. . . . 5 |- (((/) e. <.A, B>. /\ -. (/) e. <.C, D>.) -> -. <.A, B>. = <.C, D>.)
11		opprc1b 2763	. . . . 5 |- (-. A e. V <-> (/) e. <.A, B>.)
12		opprc1b 2763	. . . . . . 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 2467	. . . . 5 |- (-. A e. V -> <.A, B>. = {(/), {B}})
18		opprc1 2467	. . . . 5 |- (-. C e. V -> <.C, D>. = {(/), {D}})
19	17, 18	eqeqan12d 1466	. . . 4 |- ((-. A e. V /\ -. C e. V) -> (<.A, B>. = <.C, D>. <-> {(/), {B}} = {(/), {D}}))
20		snex 2718	. . . . . 6 |- {B} e. V
21		snex 2718	. . . . . 6 |- {D} e. V
22	20, 21	preqr2 2452	. . . . 5 |- ({(/), {B}} = {(/), {D}} -> {B} = {D})
23	5	sneqr 2447	. . . . 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 476	. 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 1099   e. wcel 1105  Vcvv 1786  (/)c0 2251  {csn 2380  {cpr 2381  <.cop 2382

This theorem is referenced by:  moop2 2766  funsn 3484

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387

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