Theorem onintss 4145

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 3661	. 2 |- ( ( A e. On /\ ps ) -> |^| { x e. On | ph } C_ A )
3	2	ex 113	1 |- ( A e. On -> ( ps -> |^| { x e. On | ph } C_ A ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   {crab 2352   C_ wss 2973   |^|cint 3636   Oncon0 4118

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-in 2979  df-ss 2986  df-int 3637

This theorem is referenced by: (None)