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Theorem onintss 6229
Description: If a property is true for an ordinal number, then the minimum ordinal number for which it is true is smaller or equal. Theorem Schema 61 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)
Hypothesis
Ref Expression
onintss.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
onintss (𝐴 ∈ On → (𝜓 {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem onintss
StepHypRef Expression
1 onintss.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
21intminss 4889 . 2 ((𝐴 ∈ On ∧ 𝜓) → {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴)
32ex 416 1 (𝐴 ∈ On → (𝜓 {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  {crab 3137  wss 3919   cint 4863  Oncon0 6179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-in 3926  df-ss 3936  df-int 4864
This theorem is referenced by:  rankval3b  9248  cardne  9387  noextenddif  33202
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