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Theorem intminss 3696
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 2762 . 2 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
3 intss1 3686 . 2 (𝐴 ∈ {𝑥𝐵𝜑} → {𝑥𝐵𝜑} ⊆ 𝐴)
42, 3sylbir 133 1 ((𝐴𝐵𝜓) → {𝑥𝐵𝜑} ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wcel 1436  {crab 2359  wss 2988   cint 3671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rab 2364  df-v 2617  df-in 2994  df-ss 3001  df-int 3672
This theorem is referenced by:  onintss  4191  cardonle  6759
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