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Theorem nepss 28920
 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

Proof of Theorem nepss
StepHypRef Expression
1 nne 2668 . . . . . 6
2 neeq1 2748 . . . . . . 7
32biimprcd 225 . . . . . 6
41, 3syl5bi 217 . . . . 5
54orrd 378 . . . 4
6 inss1 3723 . . . . . 6
76jctl 541 . . . . 5
8 inss2 3724 . . . . . 6
98jctl 541 . . . . 5
107, 9orim12i 516 . . . 4
115, 10syl 16 . . 3
12 inidm 3712 . . . . . . 7
13 ineq2 3699 . . . . . . 7
1412, 13syl5reqr 2523 . . . . . 6
1514necon3i 2707 . . . . 5
1615adantl 466 . . . 4
17 ineq1 3698 . . . . . . 7
18 inidm 3712 . . . . . . 7
1917, 18syl6eq 2524 . . . . . 6
2019necon3i 2707 . . . . 5
2120adantl 466 . . . 4
2216, 21jaoi 379 . . 3
2311, 22impbii 188 . 2
24 df-pss 3497 . . 3
25 df-pss 3497 . . 3
2624, 25orbi12i 521 . 2
2723, 26bitr4i 252 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 184   wo 368   wa 369   wceq 1379   wne 2662   cin 3480   wss 3481   wpss 3482 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3120  df-in 3488  df-ss 3495  df-pss 3497 This theorem is referenced by: (None)
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