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Theorem unssd 3176
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 3174 . . 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 2997  wss 2999
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012
This theorem is referenced by:  tpssi  3603  casef  6779  un0addcl  8706  un0mulcl  8707  fzosplit  9588  fzouzsplit  9590  strleund  11581  bj-omtrans  11851
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