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Theorem onintss 4441
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
StepHypRef Expression
1 onintss.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21intminss 3889 . 2  |-  ( ( A  e.  On  /\  ps )  ->  |^| { x  e.  On  |  ph }  C_  A )
32ex 423 1  |-  ( A  e.  On  ->  ( ps  ->  |^| { x  e.  On  |  ph }  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1685   {crab 2548    C_ wss 3153   |^|cint 3863   Oncon0 4391
This theorem is referenced by:  rankval3b  7494  cardne  7594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-rab 2553  df-v 2791  df-in 3160  df-ss 3167  df-int 3864
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