New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  disjpss GIF version

Theorem disjpss 3601
 Description: A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
disjpss (((AB) = B) → A ⊊ (AB))

Proof of Theorem disjpss
StepHypRef Expression
1 ssid 3290 . . . . . . . 8 B B
21biantru 491 . . . . . . 7 (B A ↔ (B A B B))
3 ssin 3477 . . . . . . 7 ((B A B B) ↔ B (AB))
42, 3bitri 240 . . . . . 6 (B AB (AB))
5 sseq2 3293 . . . . . 6 ((AB) = → (B (AB) ↔ B ))
64, 5syl5bb 248 . . . . 5 ((AB) = → (B AB ))
7 ss0 3581 . . . . 5 (B B = )
86, 7syl6bi 219 . . . 4 ((AB) = → (B AB = ))
98necon3ad 2552 . . 3 ((AB) = → (B → ¬ B A))
109imp 418 . 2 (((AB) = B) → ¬ B A)
11 nsspssun 3488 . . 3 B AA ⊊ (BA))
12 uncom 3408 . . . 4 (BA) = (AB)
1312psseq2i 3359 . . 3 (A ⊊ (BA) ↔ A ⊊ (AB))
1411, 13bitri 240 . 2 B AA ⊊ (AB))
1510, 14sylib 188 1 (((AB) = B) → A ⊊ (AB))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2516   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257   ⊊ wpss 3258  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-pss 3261  df-nul 3551 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator