Theorem prneimg 3894

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 3893	. . . . 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 832	. . . . . 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 531	. . . . . . 7 |- ( ( ( ( A = C \/ A = D ) /\ ( A = C \/ B = C ) ) /\ ( ( B = D \/ A = D ) /\ ( B = D \/ B = C ) ) ) -> ( A = C \/ A = D ) )
4		pm1.4 739	. . . . . . . 8 |- ( ( B = D \/ B = C ) -> ( B = C \/ B = D ) )
5	4	ad2antll 495	. . . . . . 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 306	. . . . . 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 121	. . . . 5 |- ( ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) -> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) )
8	1, 7	biimtrdi 163	. . . 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 793	. . . . . 6 |- ( ( A = C \/ A = D ) -> -. ( -. A = C /\ -. A = D ) )
10		df-ne 2421	. . . . . . 7 |- ( A =/= C <-> -. A = C )
11		df-ne 2421	. . . . . . 7 |- ( A =/= D <-> -. A = D )
12	10, 11	anbi12i 464	. . . . . 6 |- ( ( A =/= C /\ A =/= D ) <-> ( -. A = C /\ -. A = D ) )
13	9, 12	sylnibr 688	. . . . 5 |- ( ( A = C \/ A = D ) -> -. ( A =/= C /\ A =/= D ) )
14		oranim 793	. . . . . 6 |- ( ( B = C \/ B = D ) -> -. ( -. B = C /\ -. B = D ) )
15		df-ne 2421	. . . . . . 7 |- ( B =/= C <-> -. B = C )
16		df-ne 2421	. . . . . . 7 |- ( B =/= D <-> -. B = D )
17	15, 16	anbi12i 464	. . . . . 6 |- ( ( B =/= C /\ B =/= D ) <-> ( -. B = C /\ -. B = D ) )
18	14, 17	sylnibr 688	. . . . 5 |- ( ( B = C \/ B = D ) -> -. ( B =/= C /\ B =/= D ) )
19	13, 18	anim12i 338	. . . 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		pm4.56 792	. . 3 |- ( ( -. ( A =/= C /\ A =/= D ) /\ -. ( B =/= C /\ B =/= D ) ) <-> -. ( ( A =/= C /\ A =/= D ) \/ ( B =/= C /\ B =/= D ) ) )
22	20, 21	imbitrdi 161	. 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 2477	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 104   \/ wo 720   = wceq 1402   e. wcel 2209   =/= wne 2420   {cpr 3706

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712

This theorem is referenced by:  umgrvad2edg  16366  usgrexmpldifpr  16404