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Theorem disj4 4368
 Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 21-Mar-2004.)
Assertion
Ref Expression
disj4 ((𝐴𝐵) = ∅ ↔ ¬ (𝐴𝐵) ⊊ 𝐴)

Proof of Theorem disj4
StepHypRef Expression
1 disj3 4363 . 2 ((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))
2 eqcom 2805 . 2 (𝐴 = (𝐴𝐵) ↔ (𝐴𝐵) = 𝐴)
3 difss 4061 . . . 4 (𝐴𝐵) ⊆ 𝐴
4 dfpss2 4015 . . . 4 ((𝐴𝐵) ⊊ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ ¬ (𝐴𝐵) = 𝐴))
53, 4mpbiran 708 . . 3 ((𝐴𝐵) ⊊ 𝐴 ↔ ¬ (𝐴𝐵) = 𝐴)
65con2bii 361 . 2 ((𝐴𝐵) = 𝐴 ↔ ¬ (𝐴𝐵) ⊊ 𝐴)
71, 2, 63bitri 300 1 ((𝐴𝐵) = ∅ ↔ ¬ (𝐴𝐵) ⊊ 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538   ∖ cdif 3879   ∩ cin 3881   ⊆ wss 3882   ⊊ wpss 3883  ∅c0 4245 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 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-v 3443  df-dif 3885  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4246 This theorem is referenced by:  marypha1lem  8896  infeq5i  9098  wilthlem2  25695  topdifinffinlem  34831
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