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Theorem ssdifeq0 4423
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 4158 . . 3 (𝐴𝐴) = 𝐴
2 ssdifin0 4422 . . 3 (𝐴 ⊆ (𝐵𝐴) → (𝐴𝐴) = ∅)
31, 2eqtr3id 2790 . 2 (𝐴 ⊆ (𝐵𝐴) → 𝐴 = ∅)
4 0ss 4336 . . 3 ∅ ⊆ (𝐵 ∖ ∅)
5 id 22 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
6 difeq2 4057 . . . 4 (𝐴 = ∅ → (𝐵𝐴) = (𝐵 ∖ ∅))
75, 6sseq12d 3959 . . 3 (𝐴 = ∅ → (𝐴 ⊆ (𝐵𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅)))
84, 7mpbiri 259 . 2 (𝐴 = ∅ → 𝐴 ⊆ (𝐵𝐴))
93, 8impbii 208 1 (𝐴 ⊆ (𝐵𝐴) ↔ 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1539  cdif 3889  cin 3891  wss 3892  c0 4262
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3287  df-v 3439  df-dif 3895  df-in 3899  df-ss 3909  df-nul 4263
This theorem is referenced by:  disjdifprg  30955  measxun2  32219  measssd  32224  pmeasmono  32332
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