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Theorem onnminsb 4725
Description: An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set.  ps is the wff resulting from the substitution of  A for  x in wff  ph. (Contributed by NM, 9-Nov-2003.)
Hypothesis
Ref Expression
onnminsb.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
onnminsb  |-  ( A  e.  On  ->  ( A  e.  |^| { x  e.  On  |  ph }  ->  -.  ps ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem onnminsb
StepHypRef Expression
1 onnminsb.1 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21elrab 3036 . . . 4  |-  ( A  e.  { x  e.  On  |  ph }  <->  ( A  e.  On  /\  ps ) )
3 ssrab2 3372 . . . . 5  |-  { x  e.  On  |  ph }  C_  On
4 onnmin 4724 . . . . 5  |-  ( ( { x  e.  On  |  ph }  C_  On  /\  A  e.  { x  e.  On  |  ph }
)  ->  -.  A  e.  |^| { x  e.  On  |  ph }
)
53, 4mpan 652 . . . 4  |-  ( A  e.  { x  e.  On  |  ph }  ->  -.  A  e.  |^| { x  e.  On  |  ph } )
62, 5sylbir 205 . . 3  |-  ( ( A  e.  On  /\  ps )  ->  -.  A  e.  |^| { x  e.  On  |  ph }
)
76ex 424 . 2  |-  ( A  e.  On  ->  ( ps  ->  -.  A  e.  |^|
{ x  e.  On  |  ph } ) )
87con2d 109 1  |-  ( A  e.  On  ->  ( A  e.  |^| { x  e.  On  |  ph }  ->  -.  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2654    C_ wss 3264   |^|cint 3993   Oncon0 4523
This theorem is referenced by:  onminex  4728  oawordeulem  6734  oeeulem  6781  nnawordex  6817  tcrank  7742  alephnbtwn  7886  cardaleph  7904  cardmin  8373  sltval2  25335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527
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