Theorem opthreg 4654

Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4635 (via the preleq 4653 step). See df-op 3678 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 3777	. . . 4 |- A e. { A , B }
3		preleq.3	. . . . 5 |- C e. _V
4	3	prid1 3777	. . . 4 |- C e. { C , D }
5		preleq.2	. . . . . 6 |- B e. _V
6		prexg 4301	. . . . . 6 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
7	1, 5, 6	mp2an 426	. . . . 5 |- { A , B } e. _V
8		preleq.4	. . . . . 6 |- D e. _V
9		prexg 4301	. . . . . 6 |- ( ( C e. _V /\ D e. _V ) -> { C , D } e. _V )
10	3, 8, 9	mp2an 426	. . . . 5 |- { C , D } e. _V
11	1, 7, 3, 10	preleq 4653	. . . 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 436	. . 3 |- ( { A , { A , B } } = { C , { C , D } } -> ( A = C /\ { A , B } = { C , D } ) )
13		preq1 3748	. . . . . 6 |- ( A = C -> { A , B } = { C , B } )
14	13	eqeq1d 2240	. . . . 5 |- ( A = C -> ( { A , B } = { C , D } <-> { C , B } = { C , D } ) )
15	5, 8	preqr2 3852	. . . . 5 |- ( { C , B } = { C , D } -> B = D )
16	14, 15	biimtrdi 163	. . . 4 |- ( A = C -> ( { A , B } = { C , D } -> B = D ) )
17	16	imdistani 445	. . 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 3748	. . . 4 |- ( A = C -> { A , { A , B } } = { C , { A , B } } )
20	19	adantr 276	. . 3 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { A , B } } )
21		preq12 3750	. . . 4 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
22	21	preq2d 3755	. . 3 |- ( ( A = C /\ B = D ) -> { C , { A , B } } = { C , { C , D } } )
23	20, 22	eqtrd 2264	. 2 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { C , D } } )
24	18, 23	impbii 126	1 |- ( { A , { A , B } } = { C , { C , D } } <-> ( A = C /\ B = D ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   {cpr 3670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pr 4299  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-dif 3202  df-un 3204  df-sn 3675  df-pr 3676

This theorem is referenced by: (None)