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Theorem unssd 3303
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 3301 . . 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 3119  wss 3121
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134
This theorem is referenced by:  tpssi  3746  casef  7065  un0addcl  9168  un0mulcl  9169  fzosplit  10133  fzouzsplit  10135  exmidunben  12381  strleund  12506  fsumcncntop  13350  bj-charfun  13842  bj-omtrans  13991
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