Theorem opthreg 4345

Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4326 (via the preleq 4344 step). See df-op 3440 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 3531	. . . 4 |- A e. { A , B }
3		preleq.3	. . . . 5 |- C e. _V
4	3	prid1 3531	. . . 4 |- C e. { C , D }
5		preleq.2	. . . . . 6 |- B e. _V
6		prexg 4012	. . . . . 6 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
7	1, 5, 6	mp2an 417	. . . . 5 |- { A , B } e. _V
8		preleq.4	. . . . . 6 |- D e. _V
9		prexg 4012	. . . . . 6 |- ( ( C e. _V /\ D e. _V ) -> { C , D } e. _V )
10	3, 8, 9	mp2an 417	. . . . 5 |- { C , D } e. _V
11	1, 7, 3, 10	preleq 4344	. . . 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 427	. . 3 |- ( { A , { A , B } } = { C , { C , D } } -> ( A = C /\ { A , B } = { C , D } ) )
13		preq1 3502	. . . . . 6 |- ( A = C -> { A , B } = { C , B } )
14	13	eqeq1d 2093	. . . . 5 |- ( A = C -> ( { A , B } = { C , D } <-> { C , B } = { C , D } ) )
15	5, 8	preqr2 3596	. . . . 5 |- ( { C , B } = { C , D } -> B = D )
16	14, 15	syl6bi 161	. . . 4 |- ( A = C -> ( { A , B } = { C , D } -> B = D ) )
17	16	imdistani 434	. . 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 3502	. . . 4 |- ( A = C -> { A , { A , B } } = { C , { A , B } } )
20	19	adantr 270	. . 3 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { A , B } } )
21		preq12 3504	. . . 4 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
22	21	preq2d 3509	. . 3 |- ( ( A = C /\ B = D ) -> { C , { A , B } } = { C , { C , D } } )
23	20, 22	eqtrd 2117	. 2 |- ( ( A = C /\ B = D ) -> { A , { A , B } } = { C , { C , D } } )
24	18, 23	impbii 124	1 |- ( { A , { A , B } } = { C , { C , D } } <-> ( A = C /\ B = D ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1287   e. wcel 1436   _Vcvv 2615   {cpr 3432

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pr 4010  ax-setind 4326

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-dif 2990  df-un 2992  df-sn 3437  df-pr 3438

This theorem is referenced by: (None)