Theorem nepss 31599

Description: Two classes are inequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)

Assertion

Ref	Expression
nepss	|- ( A =/= B <-> ( ( A i^i B ) C. A \/ ( A i^i B ) C. B ) )

Proof of Theorem nepss

Step	Hyp	Ref	Expression
1		nne 2798	. . . . . 6 |- ( -. ( A i^i B ) =/= A <-> ( A i^i B ) = A )
2		neeq1 2856	. . . . . . 7 |- ( ( A i^i B ) = A -> ( ( A i^i B ) =/= B <-> A =/= B ) )
3	2	biimprcd 240	. . . . . 6 |- ( A =/= B -> ( ( A i^i B ) = A -> ( A i^i B ) =/= B ) )
4	1, 3	syl5bi 232	. . . . 5 |- ( A =/= B -> ( -. ( A i^i B ) =/= A -> ( A i^i B ) =/= B ) )
5	4	orrd 393	. . . 4 |- ( A =/= B -> ( ( A i^i B ) =/= A \/ ( A i^i B ) =/= B ) )
6		inss1 3833	. . . . . 6 |- ( A i^i B ) C_ A
7	6	jctl 564	. . . . 5 |- ( ( A i^i B ) =/= A -> ( ( A i^i B ) C_ A /\ ( A i^i B ) =/= A ) )
8		inss2 3834	. . . . . 6 |- ( A i^i B ) C_ B
9	8	jctl 564	. . . . 5 |- ( ( A i^i B ) =/= B -> ( ( A i^i B ) C_ B /\ ( A i^i B ) =/= B ) )
10	7, 9	orim12i 538	. . . 4 |- ( ( ( A i^i B ) =/= A \/ ( A i^i B ) =/= B ) -> ( ( ( A i^i B ) C_ A /\ ( A i^i B ) =/= A ) \/ ( ( A i^i B ) C_ B /\ ( A i^i B ) =/= B ) ) )
11	5, 10	syl 17	. . 3 |- ( A =/= B -> ( ( ( A i^i B ) C_ A /\ ( A i^i B ) =/= A ) \/ ( ( A i^i B ) C_ B /\ ( A i^i B ) =/= B ) ) )
12		inidm 3822	. . . . . . 7 |- ( A i^i A ) = A
13		ineq2 3808	. . . . . . 7 |- ( A = B -> ( A i^i A ) = ( A i^i B ) )
14	12, 13	syl5reqr 2671	. . . . . 6 |- ( A = B -> ( A i^i B ) = A )
15	14	necon3i 2826	. . . . 5 |- ( ( A i^i B ) =/= A -> A =/= B )
16	15	adantl 482	. . . 4 |- ( ( ( A i^i B ) C_ A /\ ( A i^i B ) =/= A ) -> A =/= B )
17		ineq1 3807	. . . . . . 7 |- ( A = B -> ( A i^i B ) = ( B i^i B ) )
18		inidm 3822	. . . . . . 7 |- ( B i^i B ) = B
19	17, 18	syl6eq 2672	. . . . . 6 |- ( A = B -> ( A i^i B ) = B )
20	19	necon3i 2826	. . . . 5 |- ( ( A i^i B ) =/= B -> A =/= B )
21	20	adantl 482	. . . 4 |- ( ( ( A i^i B ) C_ B /\ ( A i^i B ) =/= B ) -> A =/= B )
22	16, 21	jaoi 394	. . 3 |- ( ( ( ( A i^i B ) C_ A /\ ( A i^i B ) =/= A ) \/ ( ( A i^i B ) C_ B /\ ( A i^i B ) =/= B ) ) -> A =/= B )
23	11, 22	impbii 199	. 2 |- ( A =/= B <-> ( ( ( A i^i B ) C_ A /\ ( A i^i B ) =/= A ) \/ ( ( A i^i B ) C_ B /\ ( A i^i B ) =/= B ) ) )
24		df-pss 3590	. . 3 |- ( ( A i^i B ) C. A <-> ( ( A i^i B ) C_ A /\ ( A i^i B ) =/= A ) )
25		df-pss 3590	. . 3 |- ( ( A i^i B ) C. B <-> ( ( A i^i B ) C_ B /\ ( A i^i B ) =/= B ) )
26	24, 25	orbi12i 543	. 2 |- ( ( ( A i^i B ) C. A \/ ( A i^i B ) C. B ) <-> ( ( ( A i^i B ) C_ A /\ ( A i^i B ) =/= A ) \/ ( ( A i^i B ) C_ B /\ ( A i^i B ) =/= B ) ) )
27	23, 26	bitr4i 267	1 |- ( A =/= B <-> ( ( A i^i B ) C. A \/ ( A i^i B ) C. B ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   =/= wne 2794   i^i cin 3573   C_ wss 3574   C. wpss 3575

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-in 3581  df-ss 3588  df-pss 3590

This theorem is referenced by: (None)