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Theorem ssini 3205
Description: An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)
Hypotheses
Ref Expression
ssini.1 𝐴𝐵
ssini.2 𝐴𝐶
Assertion
Ref Expression
ssini 𝐴 ⊆ (𝐵𝐶)

Proof of Theorem ssini
StepHypRef Expression
1 ssini.1 . . 3 𝐴𝐵
2 ssini.2 . . 3 𝐴𝐶
31, 2pm3.2i 266 . 2 (𝐴𝐵𝐴𝐶)
4 ssin 3204 . 2 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))
53, 4mpbi 143 1 𝐴 ⊆ (𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wa 102  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:  inv1  3296
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