Theorem prneimg 3566

Description: Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)

Assertion

Ref	Expression
prneimg	|- ( ( ( A e. U /\ B e. V ) /\ ( C e. X /\ D e. Y ) ) -> ( ( ( A =/= C /\ A =/= D ) \/ ( B =/= C /\ B =/= D ) ) -> { A , B } =/= { C , D } ) )

Proof of Theorem prneimg

Step	Hyp	Ref	Expression
1		preq12bg 3565	. . . . 5 |- ( ( ( A e. U /\ B e. V ) /\ ( C e. X /\ D e. Y ) ) -> ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) )
2		orddi 766	. . . . . 6 |- ( ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) <-> ( ( ( A = C \/ A = D ) /\ ( A = C \/ B = C ) ) /\ ( ( B = D \/ A = D ) /\ ( B = D \/ B = C ) ) ) )
3		simpll 495	. . . . . . 7 |- ( ( ( ( A = C \/ A = D ) /\ ( A = C \/ B = C ) ) /\ ( ( B = D \/ A = D ) /\ ( B = D \/ B = C ) ) ) -> ( A = C \/ A = D ) )
4		pm1.4 678	. . . . . . . 8 |- ( ( B = D \/ B = C ) -> ( B = C \/ B = D ) )
5	4	ad2antll 474	. . . . . . 7 |- ( ( ( ( A = C \/ A = D ) /\ ( A = C \/ B = C ) ) /\ ( ( B = D \/ A = D ) /\ ( B = D \/ B = C ) ) ) -> ( B = C \/ B = D ) )
6	3, 5	jca 300	. . . . . 6 |- ( ( ( ( A = C \/ A = D ) /\ ( A = C \/ B = C ) ) /\ ( ( B = D \/ A = D ) /\ ( B = D \/ B = C ) ) ) -> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) )
7	2, 6	sylbi 119	. . . . 5 |- ( ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) -> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) )
8	1, 7	syl6bi 161	. . . 4 |- ( ( ( A e. U /\ B e. V ) /\ ( C e. X /\ D e. Y ) ) -> ( { A , B } = { C , D } -> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) ) )
9		oranim 840	. . . . . 6 |- ( ( A = C \/ A = D ) -> -. ( -. A = C /\ -. A = D ) )
10		df-ne 2246	. . . . . . 7 |- ( A =/= C <-> -. A = C )
11		df-ne 2246	. . . . . . 7 |- ( A =/= D <-> -. A = D )
12	10, 11	anbi12i 447	. . . . . 6 |- ( ( A =/= C /\ A =/= D ) <-> ( -. A = C /\ -. A = D ) )
13	9, 12	sylnibr 634	. . . . 5 |- ( ( A = C \/ A = D ) -> -. ( A =/= C /\ A =/= D ) )
14		oranim 840	. . . . . 6 |- ( ( B = C \/ B = D ) -> -. ( -. B = C /\ -. B = D ) )
15		df-ne 2246	. . . . . . 7 |- ( B =/= C <-> -. B = C )
16		df-ne 2246	. . . . . . 7 |- ( B =/= D <-> -. B = D )
17	15, 16	anbi12i 447	. . . . . 6 |- ( ( B =/= C /\ B =/= D ) <-> ( -. B = C /\ -. B = D ) )
18	14, 17	sylnibr 634	. . . . 5 |- ( ( B = C \/ B = D ) -> -. ( B =/= C /\ B =/= D ) )
19	13, 18	anim12i 331	. . . 4 |- ( ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) -> ( -. ( A =/= C /\ A =/= D ) /\ -. ( B =/= C /\ B =/= D ) ) )
20	8, 19	syl6 33	. . 3 |- ( ( ( A e. U /\ B e. V ) /\ ( C e. X /\ D e. Y ) ) -> ( { A , B } = { C , D } -> ( -. ( A =/= C /\ A =/= D ) /\ -. ( B =/= C /\ B =/= D ) ) ) )
21		pm4.56 839	. . 3 |- ( ( -. ( A =/= C /\ A =/= D ) /\ -. ( B =/= C /\ B =/= D ) ) <-> -. ( ( A =/= C /\ A =/= D ) \/ ( B =/= C /\ B =/= D ) ) )
22	20, 21	syl6ib 159	. 2 |- ( ( ( A e. U /\ B e. V ) /\ ( C e. X /\ D e. Y ) ) -> ( { A , B } = { C , D } -> -. ( ( A =/= C /\ A =/= D ) \/ ( B =/= C /\ B =/= D ) ) ) )
23	22	necon2ad 2302	1 |- ( ( ( A e. U /\ B e. V ) /\ ( C e. X /\ D e. Y ) ) -> ( ( ( A =/= C /\ A =/= D ) \/ ( B =/= C /\ B =/= D ) ) -> { A , B } =/= { C , D } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 661   = wceq 1284   e. wcel 1433   =/= wne 2245   {cpr 3399

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405

This theorem is referenced by: (None)