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Theorem onintss 5692
 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 4438 . 2 ((𝐴 ∈ On ∧ 𝜓) → {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴)
32ex 449 1 (𝐴 ∈ On → (𝜓 {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {crab 2900   ⊆ wss 3540  ∩ cint 4410  Oncon0 5640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-int 4411 This theorem is referenced by:  rankval3b  8572  cardne  8674
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