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Theorem nsspssun 4154
 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 4070 . . . 4 𝐵 ⊆ (𝐴𝐵)
21biantrur 531 . . 3 (¬ (𝐴𝐵) ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
3 ssid 3910 . . . . 5 𝐵𝐵
43biantru 530 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐵𝐵))
5 unss 4081 . . . 4 ((𝐴𝐵𝐵𝐵) ↔ (𝐴𝐵) ⊆ 𝐵)
64, 5bitri 276 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) ⊆ 𝐵)
72, 6xchnxbir 334 . 2 𝐴𝐵 ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
8 dfpss3 3984 . 2 (𝐵 ⊊ (𝐴𝐵) ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
97, 8bitr4i 279 1 𝐴𝐵𝐵 ⊊ (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∪ cun 3857   ⊆ wss 3859   ⊊ wpss 3860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-v 3439  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876 This theorem is referenced by:  disjpss  4324  lindsenlbs  34437
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