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Theorem intssuni2m 3667
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 3665 . 2 (∃𝑥 𝑥𝐴 𝐴 𝐴)
2 uniss 3629 . 2 (𝐴𝐵 𝐴 𝐵)
31, 2sylan9ssr 2987 1 ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴) → 𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wex 1397  wcel 1409  wss 2945   cuni 3608   cint 3643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-int 3644
This theorem is referenced by:  rintm  3772  onintonm  4271
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