Theorem onintss 4422

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 3896	. 2 |- ( ( A e. On /\ ps ) -> |^| { x e. On | ph } C_ A )
3	2	ex 115	1 |- ( A e. On -> ( ps -> |^| { x e. On | ph } C_ A ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   {crab 2476   C_ wss 3154   |^|cint 3871   Oncon0 4395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-in 3160  df-ss 3167  df-int 3872

This theorem is referenced by: (None)