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Theorem ssini 3160
 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 257 . 2 (𝐴𝐵𝐴𝐶)
4 ssin 3159 . 2 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))
53, 4mpbi 133 1 𝐴 ⊆ (𝐵𝐶)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ∩ cin 2916   ⊆ wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931 This theorem is referenced by:  inv1  3253
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