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Theorem unssad 3163
Description: If (𝐴𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 3160. Partial converse of unssd 3162. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
unssad.1 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
Assertion
Ref Expression
unssad (𝜑𝐴𝐶)

Proof of Theorem unssad
StepHypRef Expression
1 unssad.1 . . 3 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
2 unss 3160 . . 3 ((𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)
31, 2sylibr 132 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
43simpld 110 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  cun 2984  wss 2986
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-un 2990  df-in 2992  df-ss 2999
This theorem is referenced by:  ersym  6237  findcard2d  6540  findcard2sd  6541  diffifi  6543
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