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Theorem unssi 3251
Description: An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
Hypotheses
Ref Expression
unssi.1 𝐴𝐶
unssi.2 𝐵𝐶
Assertion
Ref Expression
unssi (𝐴𝐵) ⊆ 𝐶

Proof of Theorem unssi
StepHypRef Expression
1 unssi.1 . . 3 𝐴𝐶
2 unssi.2 . . 3 𝐵𝐶
31, 2pm3.2i 270 . 2 (𝐴𝐶𝐵𝐶)
4 unss 3250 . 2 ((𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)
53, 4mpbi 144 1 (𝐴𝐵) ⊆ 𝐶
Colors of variables: wff set class
Syntax hints:  wa 103  cun 3069  wss 3071
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084
This theorem is referenced by:  undifabs  3439  inundifss  3440  dmrnssfld  4802  caserel  6972  ltrelxr  7832  nn0ssre  8988  nn0ssz  9079  strleun  12057
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