Theorem opeqsn 3980

Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

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		opeqsn.1	. . . 4 |- A e. _V
2		opeqsn.2	. . . 4 |- B e. _V
3	1, 2	dfop 3539	. . 3 |- <.A , B >. = { {A } , {A , B } }
4	3	eqeq1i 2044	. 2 |- ( <.A , B >. = { C } <-> { {A } , {A , B } } = { C })
5		snexgOLD 3926	. . . 4 |- (A e. _V -> {A } e. _V)
6	1, 5	ax-mp 7	. . 3 |- {A } e. _V
7		prexgOLD 3937	. . . 4 |- ((A e. _V /\ B e. _V) -> {A , B } e. _V)
8	1, 2, 7	mp2an 402	. . 3 |- {A , B } e. _V
9		opeqsn.3	. . 3 |- C e. _V
10	6, 8, 9	preqsn 3537	. 2 |- ( { {A } , {A , B } } = { C } <-> ( {A } = {A , B } /\ {A , B } = C))
11		eqcom 2039	. . . . 5 |- ( {A } = {A , B } <-> {A , B } = {A })
12	1, 2, 1	preqsn 3537	. . . . 5 |- ( {A , B } = {A } <-> (A = B /\ B = A))
13		eqcom 2039	. . . . . . 7 |- (B = A <-> A = B)
14	13	anbi2i 430	. . . . . 6 |- ((A = B /\ B = A) <-> (A = B /\ A = B))
15		anidm 376	. . . . . 6 |- ((A = B /\ A = B) <-> A = B)
16	14, 15	bitri 173	. . . . 5 |- ((A = B /\ B = A) <-> A = B)
17	11, 12, 16	3bitri 195	. . . 4 |- ( {A } = {A , B } <-> A = B)
18	17	anbi1i 431	. . 3 |- (( {A } = {A , B } /\ {A , B } = C) <-> (A = B /\ {A , B } = C))
19		dfsn2 3381	. . . . . . 7 |- {A } = {A , A }
20		preq2 3439	. . . . . . 7 |- (A = B -> {A , A } = {A , B })
21	19, 20	syl5req 2082	. . . . . 6 |- (A = B -> {A , B } = {A })
22	21	eqeq1d 2045	. . . . 5 |- (A = B -> ( {A , B } = C <-> {A } = C))
23		eqcom 2039	. . . . 5 |- ( {A } = C <-> C = {A })
24	22, 23	syl6bb 185	. . . 4 |- (A = B -> ( {A , B } = C <-> C = {A }))
25	24	pm5.32i 427	. . 3 |- ((A = B /\ {A , B } = C) <-> (A = B /\ C = {A }))
26	18, 25	bitri 173	. 2 |- (( {A } = {A , B } /\ {A , B } = C) <-> (A = B /\ C = {A }))
27	4, 10, 26	3bitri 195	1 |- ( <.A , B >. = { C } <-> (A = B /\ C = {A }))

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1242   e. wcel 1390   _Vcvv 2551   {csn 3367   {cpr 3368   <.cop 3370

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376

This theorem is referenced by:  relop  4429