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Theorem intssuni2m 3639
 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 3637 . 2 (∃𝑥 𝑥𝐴 𝐴 𝐴)
2 uniss 3601 . 2 (𝐴𝐵 𝐴 𝐵)
31, 2sylan9ssr 2959 1 ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴) → 𝐴 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1381   ∈ wcel 1393   ⊆ wss 2917  ∪ cuni 3580  ∩ cint 3615 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-int 3616 This theorem is referenced by:  rintm  3744  onintonm  4243
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