Theorem opthpr 4384

Description: An unordered pair has the ordered pair property (compare opth 4945) under certain conditions. (Contributed by NM, 27-Mar-2007.)

Hypotheses

Ref	Expression
preqr1.a	|- A e. _V
preqr1.b	|- B e. _V
preq12b.c	|- C e. _V
preq12b.d	|- 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		preqr1.a	. . 3 |- A e. _V
2		preqr1.b	. . 3 |- B e. _V
3		preq12b.c	. . 3 |- C e. _V
4		preq12b.d	. . 3 |- D e. _V
5	1, 2, 3, 4	preq12b 4382	. 2 |- ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
6		idd 24	. . . 4 |- ( A =/= D -> ( ( A = C /\ B = D ) -> ( A = C /\ B = D ) ) )
7		df-ne 2795	. . . . . 6 |- ( A =/= D <-> -. A = D )
8		pm2.21 120	. . . . . 6 |- ( -. A = D -> ( A = D -> ( B = C -> ( A = C /\ B = D ) ) ) )
9	7, 8	sylbi 207	. . . . 5 |- ( A =/= D -> ( A = D -> ( B = C -> ( A = C /\ B = D ) ) ) )
10	9	impd 447	. . . 4 |- ( A =/= D -> ( ( A = D /\ B = C ) -> ( A = C /\ B = D ) ) )
11	6, 10	jaod 395	. . 3 |- ( A =/= D -> ( ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) -> ( A = C /\ B = D ) ) )
12		orc 400	. . 3 |- ( ( A = C /\ B = D ) -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
13	11, 12	impbid1 215	. 2 |- ( A =/= D -> ( ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) <-> ( A = C /\ B = D ) ) )
14	5, 13	syl5bb 272	1 |- ( A =/= D -> ( { A , B } = { C , D } <-> ( A = C /\ B = D ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  brdom7disj  9353  brdom6disj  9354