Theorem onintss 2266

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} (_ A))

Distinct variable group(s):   ps,x   x,A

Proof of Theorem onintss

Step	Hyp	Ref	Expression
1		onintss.1	. . . 4 |- (x = A -> (ph <-> ps))
2	1	elrab 1422	. . 3 |- (A e. {x e. On | ph} <-> (A e. On /\ ps))
3		intss1 1979	. . 3 |- (A e. {x e. On | ph} -> |^|{x e. On | ph} (_ A)
4	2, 3	sylbir 176	. 2 |- ((A e. On /\ ps) -> |^|{x e. On | ph} (_ A)
5	4	exp 291	1 |- (A e. On -> (ps -> |^|{x e. On | ph} (_ A))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  {crab 1204   (_ wss 1487  |^|cint 1965  Oncon0 2199

This theorem is referenced by:  rankr1 3518  rankval3 3525  oncard 3636  cardne 3637

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208  df-v 1349  df-in 1491  df-ss 1492  df-int 1966

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