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Theorem sslin 3208
 Description: Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
Assertion
Ref Expression
sslin (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Proof of Theorem sslin
StepHypRef Expression
1 ssrin 3207 . 2 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 incom 3174 . 2 (𝐶𝐴) = (𝐴𝐶)
3 incom 3174 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33sstr4g 3049 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∩ cin 2981   ⊆ wss 2982 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-in 2988  df-ss 2995 This theorem is referenced by:  ss2in  3209  difdifdirss  3343  ssres2  4686  ssrnres  4813  ioodisj  9143
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