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Theorem ssinss2d 44897
Description: Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
ssinss2d.1 (𝜑𝐵𝐶)
Assertion
Ref Expression
ssinss2d (𝜑 → (𝐴𝐵) ⊆ 𝐶)

Proof of Theorem ssinss2d
StepHypRef Expression
1 incom 4224 . 2 (𝐴𝐵) = (𝐵𝐴)
2 ssinss2d.1 . . 3 (𝜑𝐵𝐶)
32ssinss1d 44885 . 2 (𝜑 → (𝐵𝐴) ⊆ 𝐶)
41, 3eqsstrid 4051 1 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3969  wss 3970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3439  df-v 3484  df-in 3977  df-ss 3987
This theorem is referenced by:  caragenuncllem  46368
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