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Theorem ssdifeq0 4003
 Description: A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
Assertion
Ref Expression
ssdifeq0 (𝐴 ⊆ (𝐵𝐴) ↔ 𝐴 = ∅)

Proof of Theorem ssdifeq0
StepHypRef Expression
1 inidm 3784 . . 3 (𝐴𝐴) = 𝐴
2 ssdifin0 4002 . . 3 (𝐴 ⊆ (𝐵𝐴) → (𝐴𝐴) = ∅)
31, 2syl5eqr 2658 . 2 (𝐴 ⊆ (𝐵𝐴) → 𝐴 = ∅)
4 0ss 3924 . . 3 ∅ ⊆ (𝐵 ∖ ∅)
5 id 22 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
6 difeq2 3684 . . . 4 (𝐴 = ∅ → (𝐵𝐴) = (𝐵 ∖ ∅))
75, 6sseq12d 3597 . . 3 (𝐴 = ∅ → (𝐴 ⊆ (𝐵𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅)))
84, 7mpbiri 247 . 2 (𝐴 = ∅ → 𝐴 ⊆ (𝐵𝐴))
93, 8impbii 198 1 (𝐴 ⊆ (𝐵𝐴) ↔ 𝐴 = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by:  disjdifprg  28770  measxun2  29600  measssd  29605  pmeasmono  29713
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