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Theorem disj4 4286
 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 4281 . 2 ((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))
2 eqcom 2780 . 2 (𝐴 = (𝐴𝐵) ↔ (𝐴𝐵) = 𝐴)
3 difss 3993 . . . 4 (𝐴𝐵) ⊆ 𝐴
4 dfpss2 3947 . . . 4 ((𝐴𝐵) ⊊ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ ¬ (𝐴𝐵) = 𝐴))
53, 4mpbiran 697 . . 3 ((𝐴𝐵) ⊊ 𝐴 ↔ ¬ (𝐴𝐵) = 𝐴)
65con2bii 350 . 2 ((𝐴𝐵) = 𝐴 ↔ ¬ (𝐴𝐵) ⊊ 𝐴)
71, 2, 63bitri 289 1 ((𝐴𝐵) = ∅ ↔ ¬ (𝐴𝐵) ⊊ 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 198   = wceq 1508   ∖ cdif 3821   ∩ cin 3823   ⊆ wss 3824   ⊊ wpss 3825  ∅c0 4173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-v 3412  df-dif 3827  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174 This theorem is referenced by:  marypha1lem  8691  infeq5i  8892  wilthlem2  25364  topdifinffinlem  34103
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