Theorem opthreg 8141

Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 8125 (via the preleq 8140 step). See df-op 3966 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 4071	. . . 4 |- A e. { A , B }
3		preleq.3	. . . . 5 |- C e. _V
4	3	prid1 4071	. . . 4 |- C e. { C , D }
5		prex 4642	. . . . 5 |- { A , B } e. _V
6		prex 4642	. . . . 5 |- { C , D } e. _V
7	1, 5, 3, 6	preleq 8140	. . . 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 696	. . 3 |- ( { A , { A , B } } = { C , { C , D } } -> ( A = C /\ { A , B } = { C , D } ) )
9		preq1 4042	. . . . . 6 |- ( A = C -> { A , B } = { C , B } )
10	9	eqeq1d 2473	. . . . 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 4141	. . . . 5 |- ( { C , B } = { C , D } -> B = D )
14	10, 13	syl6bi 236	. . . 4 |- ( A = C -> ( { A , B } = { C , D } -> B = D ) )
15	14	imdistani 704	. . 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 4042	. . . 4 |- ( A = C -> { A , { A , B } } = { C , { A , B } } )
18	17	adantr 472	. . 3 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { A , B } } )
19		preq12 4044	. . . 4 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
20	19	preq2d 4049	. . 3 |- ( ( A = C /\ B = D ) -> { C , { A , B } } = { C , { C , D } } )
21	18, 20	eqtrd 2505	. 2 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { C , D } } )
22	16, 21	impbii 192	1 |- ( { A , { A , B } } = { C , { C , D } } <-> ( A = C /\ B = D ) )

Syntax hints:   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   _Vcvv 3031   {cpr 3961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-reg 8125

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-eprel 4750  df-fr 4798

This theorem is referenced by: (None)