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Theorem intminss 3856
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 2886 . 2 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
3 intss1 3846 . 2 (𝐴 ∈ {𝑥𝐵𝜑} → {𝑥𝐵𝜑} ⊆ 𝐴)
42, 3sylbir 134 1 ((𝐴𝐵𝜓) → {𝑥𝐵𝜑} ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wcel 2141  {crab 2452  wss 3121   cint 3831
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134  df-int 3832
This theorem is referenced by:  onintss  4375  cardonle  7164
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