Theorem opthreg 4409

Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4390 (via the preleq 4408 step). See df-op 3483 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)

Hypotheses

Ref	Expression
preleq.1	|- A e. _V
preleq.2	|- B e. _V
preleq.3	|- C e. _V
preleq.4	|- D e. _V

Assertion

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

Proof of Theorem opthreg

Step	Hyp	Ref	Expression
1		preleq.1	. . . . 5 |- A e. _V
2	1	prid1 3576	. . . 4 |- A e. { A , B }
3		preleq.3	. . . . 5 |- C e. _V
4	3	prid1 3576	. . . 4 |- C e. { C , D }
5		preleq.2	. . . . . 6 |- B e. _V
6		prexg 4071	. . . . . 6 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
7	1, 5, 6	mp2an 420	. . . . 5 |- { A , B } e. _V
8		preleq.4	. . . . . 6 |- D e. _V
9		prexg 4071	. . . . . 6 |- ( ( C e. _V /\ D e. _V ) -> { C , D } e. _V )
10	3, 8, 9	mp2an 420	. . . . 5 |- { C , D } e. _V
11	1, 7, 3, 10	preleq 4408	. . . 4 |- ( ( ( A e. { A , B } /\ C e. { C , D } ) /\ { A , { A , B } } = { C , { C , D } } ) -> ( A = C /\ { A , B } = { C , D } ) )
12	2, 4, 11	mpanl12 430	. . 3 |- ( { A , { A , B } } = { C , { C , D } } -> ( A = C /\ { A , B } = { C , D } ) )
13		preq1 3547	. . . . . 6 |- ( A = C -> { A , B } = { C , B } )
14	13	eqeq1d 2108	. . . . 5 |- ( A = C -> ( { A , B } = { C , D } <-> { C , B } = { C , D } ) )
15	5, 8	preqr2 3643	. . . . 5 |- ( { C , B } = { C , D } -> B = D )
16	14, 15	syl6bi 162	. . . 4 |- ( A = C -> ( { A , B } = { C , D } -> B = D ) )
17	16	imdistani 437	. . 3 |- ( ( A = C /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )
18	12, 17	syl 14	. 2 |- ( { A , { A , B } } = { C , { C , D } } -> ( A = C /\ B = D ) )
19		preq1 3547	. . . 4 |- ( A = C -> { A , { A , B } } = { C , { A , B } } )
20	19	adantr 272	. . 3 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { A , B } } )
21		preq12 3549	. . . 4 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
22	21	preq2d 3554	. . 3 |- ( ( A = C /\ B = D ) -> { C , { A , B } } = { C , { C , D } } )
23	20, 22	eqtrd 2132	. 2 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { C , D } } )
24	18, 23	impbii 125	1 |- ( { A , { A , B } } = { C , { C , D } } <-> ( A = C /\ B = D ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1299   e. wcel 1448   _Vcvv 2641   {cpr 3475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pr 4069  ax-setind 4390

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-dif 3023  df-un 3025  df-sn 3480  df-pr 3481

This theorem is referenced by: (None)