Theorem opthreg 4622

Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4603 (via the preleq 4621 step). See df-op 3652 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 3749	. . . 4 |- A e. { A , B }
3		preleq.3	. . . . 5 |- C e. _V
4	3	prid1 3749	. . . 4 |- C e. { C , D }
5		preleq.2	. . . . . 6 |- B e. _V
6		prexg 4271	. . . . . 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 4271	. . . . . 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 4621	. . . 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 3720	. . . . . 6 |- ( A = C -> { A , B } = { C , B } )
14	13	eqeq1d 2216	. . . . 5 |- ( A = C -> ( { A , B } = { C , D } <-> { C , B } = { C , D } ) )
15	5, 8	preqr2 3823	. . . . 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 3720	. . . 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 3722	. . . 4 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
22	21	preq2d 3727	. . 3 |- ( ( A = C /\ B = D ) -> { C , { A , B } } = { C , { C , D } } )
23	20, 22	eqtrd 2240	. 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 1373   e. wcel 2178   _Vcvv 2776   {cpr 3644

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pr 4269  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-dif 3176  df-un 3178  df-sn 3649  df-pr 3650

This theorem is referenced by: (None)