Theorem nepss 13820

Description: Two classes are inequal iff their intersection is a proper subset of one of them.

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		neeq1 2024	. . . . . . 7 |- ((A i^i B) = A -> ((A i^i B) =/= B <-> A =/= B))
2	1	biimprcd 173	. . . . . 6 |- (A =/= B -> ((A i^i B) = A -> (A i^i B) =/= B))
3		nne 2021	. . . . . 6 |- (-. (A i^i B) =/= A <-> (A i^i B) = A)
4	2, 3	syl5ib 223	. . . . 5 |- (A =/= B -> (-. (A i^i B) =/= A -> (A i^i B) =/= B))
5	4	orrd 250	. . . 4 |- (A =/= B -> ((A i^i B) =/= A \/ (A i^i B) =/= B))
6		inss1 2812	. . . . . 6 |- (A i^i B) C_ A
7	6	jctl 314	. . . . 5 |- ((A i^i B) =/= A -> ((A i^i B) C_ A /\ (A i^i B) =/= A))
8		inss2 2813	. . . . . 6 |- (A i^i B) C_ B
9	8	jctl 314	. . . . 5 |- ((A i^i B) =/= B -> ((A i^i B) C_ B /\ (A i^i B) =/= B))
10	7, 9	orim12i 363	. . . 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 12	. . 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		ineq2 2790	. . . . . . 7 |- (A = B -> (A i^i A) = (A i^i B))
13		inidm 2803	. . . . . . 7 |- (A i^i A) = A
14	12, 13	syl5reqr 1943	. . . . . 6 |- (A = B -> (A i^i B) = A)
15	14	necon3i 2042	. . . . 5 |- ((A i^i B) =/= A -> A =/= B)
16	15	adantl 424	. . . 4 |- (((A i^i B) C_ A /\ (A i^i B) =/= A) -> A =/= B)
17		ineq1 2789	. . . . . . 7 |- (A = B -> (A i^i B) = (B i^i B))
18		inidm 2803	. . . . . . 7 |- (B i^i B) = B
19	17, 18	syl6eq 1944	. . . . . 6 |- (A = B -> (A i^i B) = B)
20	19	necon3i 2042	. . . . 5 |- ((A i^i B) =/= B -> A =/= B)
21	20	adantl 424	. . . 4 |- (((A i^i B) C_ B /\ (A i^i B) =/= B) -> A =/= B)
22	16, 21	jaoi 368	. . 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 174	. 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 2607	. . 3 |- ((A i^i B) C. A <-> ((A i^i B) C_ A /\ (A i^i B) =/= A))
25		df-pss 2607	. . 3 |- ((A i^i B) C. B <-> ((A i^i B) C_ B /\ (A i^i B) =/= B))
26	24, 25	orbi12i 277	. 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 193	1 |- (A =/= B <-> ((A i^i B) C. A \/ (A i^i B) C. B))

Syntax hints:  -. wn 2   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   =/= wne 2017   i^i cin 2592   C_ wss 2593   C. wpss 2594

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-in 2603  df-ss 2605  df-pss 2607

Copyright terms: Public domain