Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nsspssun Structured version   Visualization version   GIF version

Theorem nsspssun 4164
 Description: Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
nsspssun 𝐴𝐵𝐵 ⊊ (𝐴𝐵))

Proof of Theorem nsspssun
StepHypRef Expression
1 ssun2 4080 . . . 4 𝐵 ⊆ (𝐴𝐵)
21biantrur 534 . . 3 (¬ (𝐴𝐵) ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
3 ssid 3916 . . . . 5 𝐵𝐵
43biantru 533 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐵𝐵))
5 unss 4091 . . . 4 ((𝐴𝐵𝐵𝐵) ↔ (𝐴𝐵) ⊆ 𝐵)
64, 5bitri 278 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) ⊆ 𝐵)
72, 6xchnxbir 336 . 2 𝐴𝐵 ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
8 dfpss3 3994 . 2 (𝐵 ⊊ (𝐴𝐵) ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
97, 8bitr4i 281 1 𝐴𝐵𝐵 ⊊ (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∪ cun 3858   ⊆ wss 3860   ⊊ wpss 3861 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879 This theorem is referenced by:  disjpss  4360  lindsenlbs  35358
 Copyright terms: Public domain W3C validator