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Theorem nsspssun 4268
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 4179 . . . 4 𝐵 ⊆ (𝐴𝐵)
21biantrur 530 . . 3 (¬ (𝐴𝐵) ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
3 ssid 4006 . . . . 5 𝐵𝐵
43biantru 529 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐵𝐵))
5 unss 4190 . . . 4 ((𝐴𝐵𝐵𝐵) ↔ (𝐴𝐵) ⊆ 𝐵)
64, 5bitri 275 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) ⊆ 𝐵)
72, 6xchnxbir 333 . 2 𝐴𝐵 ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
8 dfpss3 4089 . 2 (𝐵 ⊊ (𝐴𝐵) ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
97, 8bitr4i 278 1 𝐴𝐵𝐵 ⊊ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  cun 3949  wss 3951  wpss 3952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-un 3956  df-ss 3968  df-pss 3971
This theorem is referenced by:  disjpss  4461  lindsenlbs  37622
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