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Theorem onnminsb 4611
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 2936 . . . 4  |-  ( A  e.  { x  e.  On  |  ph }  <->  ( A  e.  On  /\  ps ) )
3 ssrab2 3271 . . . . 5  |-  { x  e.  On  |  ph }  C_  On
4 onnmin 4610 . . . . 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 1632    e. wcel 1696   {crab 2560    C_ wss 3165   |^|cint 3878   Oncon0 4408
This theorem is referenced by:  onminex  4614  oawordeulem  6568  oeeulem  6615  nnawordex  6651  tcrank  7570  alephnbtwn  7714  cardaleph  7732  cardmin  8202  sltval2  24381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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