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Theorem ssdisjd 46153
Description: Subset preserves disjointness. Deduction form of ssdisj 4393. (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 4169 . 2 (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))
3 ssdisjd.2 . 2 (𝜑 → (𝐵𝐶) = ∅)
4 sseq0 4333 . 2 (((𝐴𝐶) ⊆ (𝐵𝐶) ∧ (𝐵𝐶) = ∅) → (𝐴𝐶) = ∅)
52, 3, 4syl2anc 584 1 (𝜑 → (𝐴𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  cin 3886  wss 3887  c0 4256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257
This theorem is referenced by:  predisj  46156  iccdisj2  46191  sepdisj  46218  seposep  46219
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