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Theorem ssdisjdr 48728
Description: Subset preserves disjointness. Deduction form of ssdisj 4460. Alternatively this could be proved with ineqcom 4210 in tandem with ssdisjd 48727. (Contributed by Zhi Wang, 7-Sep-2024.)
Hypotheses
Ref Expression
ssdisjd.1 (𝜑𝐴𝐵)
ssdisjdr.2 (𝜑 → (𝐶𝐵) = ∅)
Assertion
Ref Expression
ssdisjdr (𝜑 → (𝐶𝐴) = ∅)

Proof of Theorem ssdisjdr
StepHypRef Expression
1 ssdisjd.1 . . 3 (𝜑𝐴𝐵)
2 sslin 4243 . . 3 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
31, 2syl 17 . 2 (𝜑 → (𝐶𝐴) ⊆ (𝐶𝐵))
4 ssdisjdr.2 . 2 (𝜑 → (𝐶𝐵) = ∅)
5 sseq0 4403 . 2 (((𝐶𝐴) ⊆ (𝐶𝐵) ∧ (𝐶𝐵) = ∅) → (𝐶𝐴) = ∅)
63, 4, 5syl2anc 584 1 (𝜑 → (𝐶𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cin 3950  wss 3951  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334
This theorem is referenced by:  predisj  48730  seposep  48823
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