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Theorem ssinss2d 39542
 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 3838 . 2 (𝐴𝐵) = (𝐵𝐴)
2 ssinss2d.1 . . 3 (𝜑𝐵𝐶)
32ssinss1d 39528 . 2 (𝜑 → (𝐵𝐴) ⊆ 𝐶)
41, 3syl5eqss 3682 1 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∩ cin 3606   ⊆ wss 3607 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621 This theorem is referenced by:  caragenuncllem  41047
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