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Theorem intssuni2m 3795
 Description: Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)
Assertion
Ref Expression
intssuni2m ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴) → 𝐴 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem intssuni2m
StepHypRef Expression
1 intssunim 3793 . 2 (∃𝑥 𝑥𝐴 𝐴 𝐴)
2 uniss 3757 . 2 (𝐴𝐵 𝐴 𝐵)
31, 2sylan9ssr 3111 1 ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴) → 𝐴 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1468   ∈ wcel 1480   ⊆ wss 3071  ∪ cuni 3736  ∩ cint 3771 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-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-int 3772 This theorem is referenced by:  rintm  3905  onintonm  4433  fival  6858  fiuni  6866
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