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Theorem ssinss1d 44936
Description: Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
ssinss1d.1 (𝜑𝐴𝐶)
Assertion
Ref Expression
ssinss1d (𝜑 → (𝐴𝐵) ⊆ 𝐶)

Proof of Theorem ssinss1d
StepHypRef Expression
1 ssinss1d.1 . 2 (𝜑𝐴𝐶)
2 ssinss1 4254 . 2 (𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)
31, 2syl 17 1 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3962  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-v 3479  df-in 3970  df-ss 3980
This theorem is referenced by:  ssinss2d  44948  ovolsplit  45894  caragenuncllem  46418  carageniuncllem1  46427  ovnsplit  46554  vonvolmbllem  46566  vonvolmbl  46567
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