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Theorem ssdifeq0 4159
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 3930 . . 3 (𝐴𝐴) = 𝐴
2 ssdifin0 4158 . . 3 (𝐴 ⊆ (𝐵𝐴) → (𝐴𝐴) = ∅)
31, 2syl5eqr 2772 . 2 (𝐴 ⊆ (𝐵𝐴) → 𝐴 = ∅)
4 0ss 4080 . . 3 ∅ ⊆ (𝐵 ∖ ∅)
5 id 22 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
6 difeq2 3830 . . . 4 (𝐴 = ∅ → (𝐵𝐴) = (𝐵 ∖ ∅))
75, 6sseq12d 3740 . . 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 1596  cdif 3677  cin 3679  wss 3680  c0 4023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rab 3023  df-v 3306  df-dif 3683  df-in 3687  df-ss 3694  df-nul 4024
This theorem is referenced by:  disjdifprg  29616  measxun2  30503  measssd  30508  pmeasmono  30616
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