Theorem onintss 6017

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 → (𝜑 ↔ 𝜓))

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

Step	Hyp	Ref	Expression
1		onintss.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	intminss 4725	. 2 ⊢ ((𝐴 ∈ On ∧ 𝜓) → ∩ {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴)
3	2	ex 403	1 ⊢ (𝐴 ∈ On → (𝜓 → ∩ {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1656   ∈ wcel 2164  {crab 3121   ⊆ wss 3798  ∩ cint 4699  Oncon0 5967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-in 3805  df-ss 3812  df-int 4700

This theorem is referenced by:  rankval3b  8973  cardne  9111  noextenddif  32355