Theorem opeqsn 3549

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 3053	. . 3 |- <.A, B>. = {{A}, {A, B}}
2	1	eqeq1i 1891	. 2 |- (<.A, B>. = {C} <-> {{A}, {A, B}} = {C})
3		snex 3492	. . 3 |- {A} e. _V
4		prex 3526	. . 3 |- {A, B} e. _V
5		opeqsn.3	. . 3 |- C e. _V
6	3, 4, 5	preqsn 3157	. 2 |- ({{A}, {A, B}} = {C} <-> ({A} = {A, B} /\ {A, B} = C))
7		eqcom 1886	. . . . 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 3157	. . . . 5 |- ({A, B} = {A} <-> (A = B /\ B = A))
11		eqcom 1886	. . . . . . 7 |- (B = A <-> A = B)
12	11	anbi2i 538	. . . . . 6 |- ((A = B /\ B = A) <-> (A = B /\ A = B))
13		anidm 478	. . . . . 6 |- ((A = B /\ A = B) <-> A = B)
14	12, 13	bitri 190	. . . . 5 |- ((A = B /\ B = A) <-> A = B)
15	7, 10, 14	3bitri 194	. . . 4 |- ({A} = {A, B} <-> A = B)
16	15	anbi1i 539	. . 3 |- (({A} = {A, B} /\ {A, B} = C) <-> (A = B /\ {A, B} = C))
17		preq2 3099	. . . . . . 7 |- (A = B -> {A, A} = {A, B})
18		dfsn2 3057	. . . . . . 7 |- {A} = {A, A}
19	17, 18	syl5req 1941	. . . . . 6 |- (A = B -> {A, B} = {A})
20	19	eqeq1d 1892	. . . . 5 |- (A = B -> ({A, B} = C <-> {A} = C))
21		eqcom 1886	. . . . 5 |- ({A} = C <-> C = {A})
22	20, 21	syl6bb 595	. . . 4 |- (A = B -> ({A, B} = C <-> C = {A}))
23	22	pm5.32i 707	. . 3 |- ((A = B /\ {A, B} = C) <-> (A = B /\ C = {A}))
24	16, 23	bitri 190	. 2 |- (({A} = {A, B} /\ {A, B} = C) <-> (A = B /\ C = {A}))
25	2, 6, 24	3bitri 194	1 |- (<.A, B>. = {C} <-> (A = B /\ C = {A}))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  {csn 3044  {cpr 3045  <.cop 3046

This theorem is referenced by:  relop 4113

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053

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