Theorem opthpr 3746

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 3744	. 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 2335	. . . . . 6 |- ( A =/= D <-> -. A = D )
8		pm2.21 607	. . . . . 6 |- ( -. A = D -> ( A = D -> ( B = C -> ( A = C /\ B = D ) ) ) )
9	7, 8	sylbi 120	. . . . 5 |- ( A =/= D -> ( A = D -> ( B = C -> ( A = C /\ B = D ) ) ) )
10	9	impd 252	. . . 4 |- ( A =/= D -> ( ( A = D /\ B = C ) -> ( A = C /\ B = D ) ) )
11	6, 10	jaod 707	. . 3 |- ( A =/= D -> ( ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) -> ( A = C /\ B = D ) ) )
12		orc 702	. . 3 |- ( ( A = C /\ B = D ) -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
13	11, 12	impbid1 141	. 2 |- ( A =/= D -> ( ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) <-> ( A = C /\ B = D ) ) )
14	5, 13	syl5bb 191	1 |- ( A =/= D -> ( { A , B } = { C , D } <-> ( A = C /\ B = D ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1342   e. wcel 2135   =/= wne 2334   _Vcvv 2721   {cpr 3571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-v 2723  df-un 3115  df-sn 3576  df-pr 3577

This theorem is referenced by: (None)