Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ssdisjd Structured version   Visualization version   GIF version

Theorem ssdisjd 47763
Description: Subset preserves disjointness. Deduction form of ssdisj 4454. (Contributed by Zhi Wang, 7-Sep-2024.)
Hypotheses
Ref Expression
ssdisjd.1 (𝜑𝐴𝐵)
ssdisjd.2 (𝜑 → (𝐵𝐶) = ∅)
Assertion
Ref Expression
ssdisjd (𝜑 → (𝐴𝐶) = ∅)

Proof of Theorem ssdisjd
StepHypRef Expression
1 ssdisjd.1 . . 3 (𝜑𝐴𝐵)
21ssrind 4230 . 2 (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))
3 ssdisjd.2 . 2 (𝜑 → (𝐵𝐶) = ∅)
4 sseq0 4394 . 2 (((𝐴𝐶) ⊆ (𝐵𝐶) ∧ (𝐵𝐶) = ∅) → (𝐴𝐶) = ∅)
52, 3, 4syl2anc 583 1 (𝜑 → (𝐴𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cin 3942  wss 3943  c0 4317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-dif 3946  df-in 3950  df-ss 3960  df-nul 4318
This theorem is referenced by:  predisj  47766  iccdisj2  47801  sepdisj  47828  seposep  47829
  Copyright terms: Public domain W3C validator