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Theorem ssrind 3310
 Description: Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
ssrind.1 (𝜑𝐴𝐵)
Assertion
Ref Expression
ssrind (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem ssrind
StepHypRef Expression
1 ssrind.1 . 2 (𝜑𝐴𝐵)
2 ssrin 3308 . 2 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
31, 2syl 14 1 (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∩ cin 3077   ⊆ wss 3078 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-v 2693  df-in 3084  df-ss 3091 This theorem is referenced by:  restbasg  12412
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