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Theorem onintss 4414
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 3862 . 2  |-  ( ( A  e.  On  /\  ps )  ->  |^| { x  e.  On  |  ph }  C_  A )
32ex 425 1  |-  ( A  e.  On  ->  ( ps  ->  |^| { x  e.  On  |  ph }  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1619    e. wcel 1621   {crab 2522    C_ wss 3127   |^|cint 3836   Oncon0 4364
This theorem is referenced by:  rankval3b  7466  cardne  7566
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-rab 2527  df-v 2765  df-in 3134  df-ss 3141  df-int 3837
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