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Theorem unssbd 3121
Description: If (𝐴𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 3117. Partial converse of unssd 3119. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
unssad.1 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
Assertion
Ref Expression
unssbd (𝜑𝐵𝐶)

Proof of Theorem unssbd
StepHypRef Expression
1 unssad.1 . . 3 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
2 unss 3117 . . 3 ((𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)
31, 2sylibr 137 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
43simprd 107 1 (𝜑𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  cun 2915  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-un 2922  df-in 2924  df-ss 2931
This theorem is referenced by:  eldifpw  4208  ertr  6121  diffifi  6351
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