ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  onintss Unicode version

Theorem onintss 4373
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 3854 . 2  |-  ( ( A  e.  On  /\  ps )  ->  |^| { x  e.  On  |  ph }  C_  A )
32ex 114 1  |-  ( A  e.  On  ->  ( ps  ->  |^| { x  e.  On  |  ph }  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348    e. wcel 2141   {crab 2452    C_ wss 3121   |^|cint 3829   Oncon0 4346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134  df-int 3830
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator