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Theorem nsspssun 3488
 Description: Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
nsspssun A BB ⊊ (AB))

Proof of Theorem nsspssun
StepHypRef Expression
1 ssun2 3427 . . . 4 B (AB)
21biantrur 492 . . 3 (¬ (AB) B ↔ (B (AB) ¬ (AB) B))
3 ssid 3290 . . . . 5 B B
43biantru 491 . . . 4 (A B ↔ (A B B B))
5 unss 3437 . . . 4 ((A B B B) ↔ (AB) B)
64, 5bitri 240 . . 3 (A B ↔ (AB) B)
72, 6xchnxbir 300 . 2 A B ↔ (B (AB) ¬ (AB) B))
8 dfpss3 3355 . 2 (B ⊊ (AB) ↔ (B (AB) ¬ (AB) B))
97, 8bitr4i 243 1 A BB ⊊ (AB))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ∪ cun 3207   ⊆ wss 3257   ⊊ wpss 3258 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-ss 3259  df-pss 3261 This theorem is referenced by:  disjpss  3601
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