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Theorem onintss 5919
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 4638 . 2 ((𝐴 ∈ On ∧ 𝜓) → {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴)
32ex 397 1 (𝐴 ∈ On → (𝜓 {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  {crab 3065  wss 3724   cint 4612  Oncon0 5867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-in 3731  df-ss 3738  df-int 4613
This theorem is referenced by:  rankval3b  8854  cardne  8992  noextenddif  32159
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