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Theorem ssdisjd 48656
Description: Subset preserves disjointness. Deduction form of ssdisj 4466. (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 4252 . 2 (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))
3 ssdisjd.2 . 2 (𝜑 → (𝐵𝐶) = ∅)
4 sseq0 4409 . 2 (((𝐴𝐶) ⊆ (𝐵𝐶) ∧ (𝐵𝐶) = ∅) → (𝐴𝐶) = ∅)
52, 3, 4syl2anc 584 1 (𝜑 → (𝐴𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  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-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340
This theorem is referenced by:  predisj  48659  iccdisj2  48694  sepdisj  48721  seposep  48722
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