Theorem onintss 4480

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 3947	. 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 1395   e. wcel 2200   {crab 2512   C_ wss 3197   |^|cint 3922   Oncon0 4453

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-in 3203  df-ss 3210  df-int 3923

This theorem is referenced by: (None)