MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssdifeq0 Structured version   Visualization version   GIF version

Theorem ssdifeq0 4493
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 4235 . . 3 (𝐴𝐴) = 𝐴
2 ssdifin0 4492 . . 3 (𝐴 ⊆ (𝐵𝐴) → (𝐴𝐴) = ∅)
31, 2eqtr3id 2789 . 2 (𝐴 ⊆ (𝐵𝐴) → 𝐴 = ∅)
4 0ss 4406 . . 3 ∅ ⊆ (𝐵 ∖ ∅)
5 id 22 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
6 difeq2 4130 . . . 4 (𝐴 = ∅ → (𝐵𝐴) = (𝐵 ∖ ∅))
75, 6sseq12d 4029 . . 3 (𝐴 = ∅ → (𝐴 ⊆ (𝐵𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅)))
84, 7mpbiri 258 . 2 (𝐴 = ∅ → 𝐴 ⊆ (𝐵𝐴))
93, 8impbii 209 1 (𝐴 ⊆ (𝐵𝐴) ↔ 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  cdif 3960  cin 3962  wss 3963  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340
This theorem is referenced by:  disjdifprg  32595  measxun2  34191  measssd  34196  pmeasmono  34306
  Copyright terms: Public domain W3C validator