Theorem onintss 4127

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 3640	. 2 |- ( ( A e. On /\ ps ) -> |^| { x e. On | ph } C_ A )
3	2	ex 108	1 |- ( A e. On -> ( ps -> |^| { x e. On | ph } C_ A ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   {crab 2310   C_ wss 2917   |^|cint 3615   Oncon0 4100

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-int 3616

This theorem is referenced by: (None)