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Theorem nepss 25175
 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 2605 . . . . . 6
2 neeq1 2609 . . . . . . 7
32biimprcd 217 . . . . . 6
41, 3syl5bi 209 . . . . 5
54orrd 368 . . . 4
6 inss1 3561 . . . . . 6
76jctl 526 . . . . 5
8 inss2 3562 . . . . . 6
98jctl 526 . . . . 5
107, 9orim12i 503 . . . 4
115, 10syl 16 . . 3
12 inidm 3550 . . . . . . 7
13 ineq2 3536 . . . . . . 7
1412, 13syl5reqr 2483 . . . . . 6
1514necon3i 2643 . . . . 5
1615adantl 453 . . . 4
17 ineq1 3535 . . . . . . 7
18 inidm 3550 . . . . . . 7
1917, 18syl6eq 2484 . . . . . 6
2019necon3i 2643 . . . . 5
2120adantl 453 . . . 4
2216, 21jaoi 369 . . 3
2311, 22impbii 181 . 2
24 df-pss 3336 . . 3
25 df-pss 3336 . . 3
2624, 25orbi12i 508 . 2
2723, 26bitr4i 244 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 177   wo 358   wa 359   wceq 1652   wne 2599   cin 3319   wss 3320   wpss 3321 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-in 3327  df-ss 3334  df-pss 3336
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