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Theorem unssd 3284
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 3282 . . 3 ((𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)
43biimpi 119 . 2 ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)
51, 2, 4syl2anc 409 1 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  cun 3100  wss 3102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115
This theorem is referenced by:  tpssi  3724  casef  7035  un0addcl  9129  un0mulcl  9130  fzosplit  10086  fzouzsplit  10088  exmidunben  12251  strleund  12374  fsumcncntop  13052  bj-charfun  13479  bj-omtrans  13628
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