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Theorem intminss 4981
Description: Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)
Hypothesis
Ref Expression
intminss.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
intminss ((𝐴𝐵𝜓) → {𝑥𝐵𝜑} ⊆ 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem intminss
StepHypRef Expression
1 intminss.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
21elrab 3684 . 2 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
3 intss1 4970 . 2 (𝐴 ∈ {𝑥𝐵𝜑} → {𝑥𝐵𝜑} ⊆ 𝐴)
42, 3sylbir 234 1 ((𝐴𝐵𝜓) → {𝑥𝐵𝜑} ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  {crab 3430  wss 3949   cint 4953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-in 3956  df-ss 3966  df-int 4954
This theorem is referenced by:  onintss  6425  knatar  7371  dfttrcl2  9755  cardonle  9988  coftr  10304  wuncss  10776  ist1-3  23273  sigagenss  33801  ldgenpisyslem1  33815  dynkin  33819  fneint  35865  igenmin  37570  pclclN  39396  dfrcl2  43135
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