Theorem onintss 3713

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.

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

Proof of Theorem onintss

Step	Hyp	Ref	Expression
1		onintss.1	. . . 4 |- (x = A -> (ph <-> ps))
2	1	elrab 2414	. . 3 |- (A e. {x e. On | ph} <-> (A e. On /\ ps))
3		intss1 3231	. . 3 |- (A e. {x e. On | ph} -> |^|{x e. On | ph} C_ A)
4	2, 3	sylbir 218	. 2 |- ((A e. On /\ ps) -> |^|{x e. On | ph} C_ A)
5	4	ex 402	1 |- (A e. On -> (ps -> |^|{x e. On | ph} C_ A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {crab 2108   C_ wss 2593  |^|cint 3214  Oncon0 3657

This theorem is referenced by:  rankr1 5785  rankval3 5792  oncard 5978  cardne 5980

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rab 2112  df-v 2294  df-in 2603  df-ss 2605  df-int 3215

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