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Theorem ssdifeq0 4023
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 3800 . . 3 (𝐴𝐴) = 𝐴
2 ssdifin0 4022 . . 3 (𝐴 ⊆ (𝐵𝐴) → (𝐴𝐴) = ∅)
31, 2syl5eqr 2669 . 2 (𝐴 ⊆ (𝐵𝐴) → 𝐴 = ∅)
4 0ss 3944 . . 3 ∅ ⊆ (𝐵 ∖ ∅)
5 id 22 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
6 difeq2 3700 . . . 4 (𝐴 = ∅ → (𝐵𝐴) = (𝐵 ∖ ∅))
75, 6sseq12d 3613 . . 3 (𝐴 = ∅ → (𝐴 ⊆ (𝐵𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅)))
84, 7mpbiri 248 . 2 (𝐴 = ∅ → 𝐴 ⊆ (𝐵𝐴))
93, 8impbii 199 1 (𝐴 ⊆ (𝐵𝐴) ↔ 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  cdif 3552  cin 3554  wss 3555  c0 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892
This theorem is referenced by:  disjdifprg  29230  measxun2  30051  measssd  30056  pmeasmono  30164
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