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Theorem intssuni2m 3765
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 3763 . 2 (∃𝑥 𝑥𝐴 𝐴 𝐴)
2 uniss 3727 . 2 (𝐴𝐵 𝐴 𝐵)
31, 2sylan9ssr 3081 1 ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴) → 𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1453  wcel 1465  wss 3041   cuni 3706   cint 3741
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-int 3742
This theorem is referenced by:  rintm  3875  onintonm  4403  fival  6826  fiuni  6834
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