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Theorem intminss 3895
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 2916 . 2 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
3 intss1 3885 . 2 (𝐴 ∈ {𝑥𝐵𝜑} → {𝑥𝐵𝜑} ⊆ 𝐴)
42, 3sylbir 135 1 ((𝐴𝐵𝜓) → {𝑥𝐵𝜑} ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2164  {crab 2476  wss 3153   cint 3870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-in 3159  df-ss 3166  df-int 3871
This theorem is referenced by:  onintss  4421  cardonle  7247
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