Theorem opthpr 3813

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 3811	. 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 2377	. . . . . 6 |- ( A =/= D <-> -. A = D )
8		pm2.21 618	. . . . . 6 |- ( -. A = D -> ( A = D -> ( B = C -> ( A = C /\ B = D ) ) ) )
9	7, 8	sylbi 121	. . . . 5 |- ( A =/= D -> ( A = D -> ( B = C -> ( A = C /\ B = D ) ) ) )
10	9	impd 254	. . . 4 |- ( A =/= D -> ( ( A = D /\ B = C ) -> ( A = C /\ B = D ) ) )
11	6, 10	jaod 719	. . 3 |- ( A =/= D -> ( ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) -> ( A = C /\ B = D ) ) )
12		orc 714	. . 3 |- ( ( A = C /\ B = D ) -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
13	11, 12	impbid1 142	. 2 |- ( A =/= D -> ( ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) <-> ( A = C /\ B = D ) ) )
14	5, 13	bitrid 192	1 |- ( A =/= D -> ( { A , B } = { C , D } <-> ( A = C /\ B = D ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2176   =/= wne 2376   _Vcvv 2772   {cpr 3634

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-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640

This theorem is referenced by: (None)