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Theorem unssd 3113
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 3111 . . 3 ((A𝐶 B𝐶) ↔ (AB) ⊆ 𝐶)
43biimpi 113 . 2 ((A𝐶 B𝐶) → (AB) ⊆ 𝐶)
51, 2, 4syl2anc 391 1 (φ → (AB) ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  cun 2909  wss 2911
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925
This theorem is referenced by:  tpssi  3521  un0addcl  7951  un0mulcl  7952  fzosplit  8763  fzouzsplit  8765  bj-omtrans  9344
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