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Theorem ssini 3265
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 268 . 2 (𝐴𝐵𝐴𝐶)
4 ssin 3264 . 2 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))
53, 4mpbi 144 1 𝐴 ⊆ (𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wa 103  cin 3036  wss 3037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-in 3043  df-ss 3050
This theorem is referenced by:  inv1  3365
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