Theorem opthreg 7936

Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 7919 (via the preleq 7935 step). See df-op 3993 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 4092	. . . 4 |- A e. { A , B }
3		preleq.3	. . . . 5 |- C e. _V
4	3	prid1 4092	. . . 4 |- C e. { C , D }
5		prex 4643	. . . . 5 |- { A , B } e. _V
6		prex 4643	. . . . 5 |- { C , D } e. _V
7	1, 5, 3, 6	preleq 7935	. . . 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 4063	. . . . . 6 |- ( A = C -> { A , B } = { C , B } )
10	9	eqeq1d 2456	. . . . 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 4156	. . . . 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 4063	. . . 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 4065	. . . 4 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
20	19	preq2d 4070	. . 3 |- ( ( A = C /\ B = D ) -> { C , { A , B } } = { C , { C , D } } )
21	18, 20	eqtrd 2495	. 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 1370   e. wcel 1758   _Vcvv 3078   {cpr 3988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-reg 7919

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-eprel 4741  df-fr 4788

This theorem is referenced by: (None)