Theorem opeqsn 2879

Description: Equivalence for an ordered pair equal to a singleton.

Hypotheses

Ref	Expression
opeqsn.1	|- A e. V
opeqsn.2	|- B e. V
opeqsn.3	|- C e. V

Assertion

Ref	Expression
opeqsn	|- (<.A, B>. = {C} <-> (A = B /\ C = {A}))

Proof of Theorem opeqsn

Step	Hyp	Ref	Expression
1		df-op 2474	. . 3 |- <.A, B>. = {{A}, {A, B}}
2	1	eqeq1i 1525	. 2 |- (<.A, B>. = {C} <-> {{A}, {A, B}} = {C})
3		snex 2826	. . 3 |- {A} e. V
4		prex 2857	. . 3 |- {A, B} e. V
5		opeqsn.3	. . 3 |- C e. V
6	3, 4, 5	preqsn 2551	. 2 |- ({{A}, {A, B}} = {C} <-> ({A} = {A, B} /\ {A, B} = C))
7		eqcom 1520	. . . . 5 |- ({A} = {A, B} <-> {A, B} = {A})
8		opeqsn.1	. . . . . 6 |- A e. V
9		opeqsn.2	. . . . . 6 |- B e. V
10	8, 9, 8	preqsn 2551	. . . . 5 |- ({A, B} = {A} <-> (A = B /\ B = A))
11		eqcom 1520	. . . . . . 7 |- (B = A <-> A = B)
12	11	anbi2i 483	. . . . . 6 |- ((A = B /\ B = A) <-> (A = B /\ A = B))
13		anidm 433	. . . . . 6 |- ((A = B /\ A = B) <-> A = B)
14	12, 13	bitri 171	. . . . 5 |- ((A = B /\ B = A) <-> A = B)
15	7, 10, 14	3bitri 175	. . . 4 |- ({A} = {A, B} <-> A = B)
16	15	anbi1i 484	. . 3 |- (({A} = {A, B} /\ {A, B} = C) <-> (A = B /\ {A, B} = C))
17		preq2 2510	. . . . . . 7 |- (A = B -> {A, A} = {A, B})
18		dfsn2 2478	. . . . . . 7 |- {A} = {A, A}
19	17, 18	syl5req 1563	. . . . . 6 |- (A = B -> {A, B} = {A})
20	19	eqeq1d 1526	. . . . 5 |- (A = B -> ({A, B} = C <-> {A} = C))
21		eqcom 1520	. . . . 5 |- ({A} = C <-> C = {A})
22	20, 21	syl6bb 539	. . . 4 |- (A = B -> ({A, B} = C <-> C = {A}))
23	22	pm5.32i 648	. . 3 |- ((A = B /\ {A, B} = C) <-> (A = B /\ C = {A}))
24	16, 23	bitri 171	. 2 |- (({A} = {A, B} /\ {A, B} = C) <-> (A = B /\ C = {A}))
25	2, 6, 24	3bitri 175	1 |- (<.A, B>. = {C} <-> (A = B /\ C = {A}))

Syntax hints:   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994  Vcvv 1857  {csn 2467  {cpr 2468  <.cop 2469

This theorem is referenced by:  relop 3365

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474

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