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Theorem unssd 3160
Description: A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
unssd.1 (𝜑𝐴𝐶)
unssd.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
unssd (𝜑 → (𝐴𝐵) ⊆ 𝐶)

Proof of Theorem unssd
StepHypRef Expression
1 unssd.1 . 2 (𝜑𝐴𝐶)
2 unssd.2 . 2 (𝜑𝐵𝐶)
3 unss 3158 . . 3 ((𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)
43biimpi 118 . 2 ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)
51, 2, 4syl2anc 403 1 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  cun 2982  wss 2984
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 2614  df-un 2988  df-in 2990  df-ss 2997
This theorem is referenced by:  tpssi  3577  casef  6686  un0addcl  8598  un0mulcl  8599  fzosplit  9477  fzouzsplit  9479  bj-omtrans  11194
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