Theorem opthreg 4280

Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4262 (via the preleq 4279 step). See df-op 3384 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 3476	. . . 4 |- A e. { A , B }
3		preleq.3	. . . . 5 |- C e. _V
4	3	prid1 3476	. . . 4 |- C e. { C , D }
5		preleq.2	. . . . . 6 |- B e. _V
6		prexgOLD 3946	. . . . . 6 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
7	1, 5, 6	mp2an 402	. . . . 5 |- { A , B } e. _V
8		preleq.4	. . . . . 6 |- D e. _V
9		prexgOLD 3946	. . . . . 6 |- ( ( C e. _V /\ D e. _V ) -> { C , D } e. _V )
10	3, 8, 9	mp2an 402	. . . . 5 |- { C , D } e. _V
11	1, 7, 3, 10	preleq 4279	. . . 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 412	. . 3 |- ( { A , { A , B } } = { C , { C , D } } -> ( A = C /\ { A , B } = { C , D } ) )
13		preq1 3447	. . . . . 6 |- ( A = C -> { A , B } = { C , B } )
14	13	eqeq1d 2048	. . . . 5 |- ( A = C -> ( { A , B } = { C , D } <-> { C , B } = { C , D } ) )
15	5, 8	preqr2 3540	. . . . 5 |- ( { C , B } = { C , D } -> B = D )
16	14, 15	syl6bi 152	. . . 4 |- ( A = C -> ( { A , B } = { C , D } -> B = D ) )
17	16	imdistani 419	. . 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 3447	. . . 4 |- ( A = C -> { A , { A , B } } = { C , { A , B } } )
20	19	adantr 261	. . 3 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { A , B } } )
21		preq12 3449	. . . 4 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
22	21	preq2d 3454	. . 3 |- ( ( A = C /\ B = D ) -> { C , { A , B } } = { C , { C , D } } )
23	20, 22	eqtrd 2072	. 2 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { C , D } } )
24	18, 23	impbii 117	1 |- ( { A , { A , B } } = { C , { C , D } } <-> ( A = C /\ B = D ) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pr 3944  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-sn 3381  df-pr 3382

This theorem is referenced by: (None)