Theorem opthpr 3543

Description: A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)

Hypotheses

Ref	Expression
preq12b.1	|- A e. _V
preq12b.2	|- B e. _V
preq12b.3	|- C e. _V
preq12b.4	|- D e. _V

Assertion

Ref	Expression
opthpr	|- ( A =/= D -> ( { A , B } = { C , D } <-> ( A = C /\ B = D ) ) )

Proof of Theorem opthpr

Step	Hyp	Ref	Expression
1		preq12b.1	. . 3 |- A e. _V
2		preq12b.2	. . 3 |- B e. _V
3		preq12b.3	. . 3 |- C e. _V
4		preq12b.4	. . 3 |- D e. _V
5	1, 2, 3, 4	preq12b 3541	. 2 |- ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
6		idd 21	. . . 4 |- ( A =/= D -> ( ( A = C /\ B = D ) -> ( A = C /\ B = D ) ) )
7		df-ne 2206	. . . . . 6 |- ( A =/= D <-> -. A = D )
8		pm2.21 547	. . . . . 6 |- ( -. A = D -> ( A = D -> ( B = C -> ( A = C /\ B = D ) ) ) )
9	7, 8	sylbi 114	. . . . 5 |- ( A =/= D -> ( A = D -> ( B = C -> ( A = C /\ B = D ) ) ) )
10	9	impd 242	. . . 4 |- ( A =/= D -> ( ( A = D /\ B = C ) -> ( A = C /\ B = D ) ) )
11	6, 10	jaod 637	. . 3 |- ( A =/= D -> ( ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) -> ( A = C /\ B = D ) ) )
12		orc 633	. . 3 |- ( ( A = C /\ B = D ) -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
13	11, 12	impbid1 130	. 2 |- ( A =/= D -> ( ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) <-> ( A = C /\ B = D ) ) )
14	5, 13	syl5bb 181	1 |- ( A =/= D -> ( { A , B } = { C , D } <-> ( A = C /\ B = D ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   = wceq 1243   e. wcel 1393   =/= wne 2204   _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-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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382

This theorem is referenced by: (None)