Theorem onintss 3011

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 groups:   ps,x   x,A

Proof of Theorem onintss

Step	Hyp	Ref	Expression
1		onintss.1	. . . 4 |- (x = A -> (ph <-> ps))
2	1	elrab 1905	. . 3 |- (A e. {x e. On | ph} <-> (A e. On /\ ps))
3		intss1 2548	. . 3 |- (A e. {x e. On | ph} -> |^|{x e. On | ph} (_ A)
4	2, 3	sylbir 201	. 2 |- ((A e. On /\ ps) -> |^|{x e. On | ph} (_ A)
5	4	ex 373	1 |- (A e. On -> (ps -> |^|{x e. On | ph} (_ A))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {crab 1648   (_ wss 2047  |^|cint 2533  Oncon0 2948

This theorem is referenced by:  rankr1 4674  rankval3 4681  oncard 4829  cardne 4830

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652  df-v 1812  df-in 2051  df-ss 2053  df-int 2534

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