Theorem opthreg 8052

Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 8036 (via the preleq 8051 step). See df-op 4039 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 4140	. . . 4 |- A e. { A , B }
3		preleq.3	. . . . 5 |- C e. _V
4	3	prid1 4140	. . . 4 |- C e. { C , D }
5		prex 4698	. . . . 5 |- { A , B } e. _V
6		prex 4698	. . . . 5 |- { C , D } e. _V
7	1, 5, 3, 6	preleq 8051	. . . 4 |- ( ( ( A e. { A , B } /\ C e. { C , D } ) /\ { A , { A , B } } = { C , { C , D } } ) -> ( A = C /\ { A , B } = { C , D } ) )
8	2, 4, 7	mpanl12 682	. . 3 |- ( { A , { A , B } } = { C , { C , D } } -> ( A = C /\ { A , B } = { C , D } ) )
9		preq1 4111	. . . . . 6 |- ( A = C -> { A , B } = { C , B } )
10	9	eqeq1d 2459	. . . . 5 |- ( A = C -> ( { A , B } = { C , D } <-> { C , B } = { C , D } ) )
11		preleq.2	. . . . . 6 |- B e. _V
12		preleq.4	. . . . . 6 |- D e. _V
13	11, 12	preqr2 4207	. . . . 5 |- ( { C , B } = { C , D } -> B = D )
14	10, 13	syl6bi 228	. . . 4 |- ( A = C -> ( { A , B } = { C , D } -> B = D ) )
15	14	imdistani 690	. . 3 |- ( ( A = C /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )
16	8, 15	syl 16	. 2 |- ( { A , { A , B } } = { C , { C , D } } -> ( A = C /\ B = D ) )
17		preq1 4111	. . . 4 |- ( A = C -> { A , { A , B } } = { C , { A , B } } )
18	17	adantr 465	. . 3 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { A , B } } )
19		preq12 4113	. . . 4 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
20	19	preq2d 4118	. . 3 |- ( ( A = C /\ B = D ) -> { C , { A , B } } = { C , { C , D } } )
21	18, 20	eqtrd 2498	. 2 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { C , D } } )
22	16, 21	impbii 188	1 |- ( { A , { A , B } } = { C , { C , D } } <-> ( A = C /\ B = D ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   _Vcvv 3109   {cpr 4034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-reg 8036

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-eprel 4800  df-fr 4847

This theorem is referenced by: (None)