Theorem opthreg 8515

Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 8497 (via the preleq 8514 step). See df-op 4184 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 4297	. . . 4 |- A e. { A , B }
3		preleq.3	. . . . 5 |- C e. _V
4	3	prid1 4297	. . . 4 |- C e. { C , D }
5		prex 4909	. . . . 5 |- { A , B } e. _V
6		prex 4909	. . . . 5 |- { C , D } e. _V
7	1, 5, 3, 6	preleq 8514	. . . 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 718	. . 3 |- ( { A , { A , B } } = { C , { C , D } } -> ( A = C /\ { A , B } = { C , D } ) )
9		preq1 4268	. . . . . 6 |- ( A = C -> { A , B } = { C , B } )
10	9	eqeq1d 2624	. . . . 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 4381	. . . . 5 |- ( { C , B } = { C , D } -> B = D )
14	10, 13	syl6bi 243	. . . 4 |- ( A = C -> ( { A , B } = { C , D } -> B = D ) )
15	14	imdistani 726	. . 3 |- ( ( A = C /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )
16	8, 15	syl 17	. 2 |- ( { A , { A , B } } = { C , { C , D } } -> ( A = C /\ B = D ) )
17		preq1 4268	. . . 4 |- ( A = C -> { A , { A , B } } = { C , { A , B } } )
18	17	adantr 481	. . 3 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { A , B } } )
19		preq12 4270	. . . 4 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
20	19	preq2d 4275	. . 3 |- ( ( A = C /\ B = D ) -> { C , { A , B } } = { C , { C , D } } )
21	18, 20	eqtrd 2656	. 2 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { C , D } } )
22	16, 21	impbii 199	1 |- ( { A , { A , B } } = { C , { C , D } } <-> ( A = C /\ B = D ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-fr 5073

This theorem is referenced by: (None)