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Theorem onintss 4180
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 3687 . 2 ((𝐴 ∈ On ∧ 𝜓) → {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴)
32ex 113 1 (𝐴 ∈ On → (𝜓 {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  {crab 2357  wss 2984   cint 3662  Oncon0 4153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-v 2614  df-in 2990  df-ss 2997  df-int 3663
This theorem is referenced by: (None)
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