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Theorem onnminsb 4595
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 2923 . . . 4  |-  ( A  e.  { x  e.  On  |  ph }  <->  ( A  e.  On  /\  ps ) )
3 ssrab2 3258 . . . . 5  |-  { x  e.  On  |  ph }  C_  On
4 onnmin 4594 . . . . 5  |-  ( ( { x  e.  On  |  ph }  C_  On  /\  A  e.  { x  e.  On  |  ph }
)  ->  -.  A  e.  |^| { x  e.  On  |  ph }
)
53, 4mpan 651 . . . 4  |-  ( A  e.  { x  e.  On  |  ph }  ->  -.  A  e.  |^| { x  e.  On  |  ph } )
62, 5sylbir 204 . . 3  |-  ( ( A  e.  On  /\  ps )  ->  -.  A  e.  |^| { x  e.  On  |  ph }
)
76ex 423 . 2  |-  ( A  e.  On  ->  ( ps  ->  -.  A  e.  |^|
{ x  e.  On  |  ph } ) )
87con2d 107 1  |-  ( A  e.  On  ->  ( A  e.  |^| { x  e.  On  |  ph }  ->  -.  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   |^|cint 3862   Oncon0 4392
This theorem is referenced by:  onminex  4598  oawordeulem  6552  oeeulem  6599  nnawordex  6635  tcrank  7554  alephnbtwn  7698  cardaleph  7716  cardmin  8186  sltval2  24310
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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