Theorem onintss 5775

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 -> ( ph <-> ps ) )

Assertion

Ref	Expression
onintss	|- ( A e. On -> ( ps -> |^| { x e. On | ph } C_ A ) )

Distinct variable groups:   ps, x   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem onintss

Step	Hyp	Ref	Expression
1		onintss.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
2	1	intminss 4503	. 2 |- ( ( A e. On /\ ps ) -> |^| { x e. On | ph } C_ A )
3	2	ex 450	1 |- ( A e. On -> ( ps -> |^| { x e. On | ph } C_ A ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   |^|cint 4475   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-int 4476

This theorem is referenced by:  rankval3b  8689  cardne  8791  noextenddif  31821