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Theorem onnminsb 6624
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 3261 . . . 4  |-  ( A  e.  { x  e.  On  |  ph }  <->  ( A  e.  On  /\  ps ) )
3 ssrab2 3585 . . . . 5  |-  { x  e.  On  |  ph }  C_  On
4 onnmin 6623 . . . . 5  |-  ( ( { x  e.  On  |  ph }  C_  On  /\  A  e.  { x  e.  On  |  ph }
)  ->  -.  A  e.  |^| { x  e.  On  |  ph }
)
53, 4mpan 670 . . . 4  |-  ( A  e.  { x  e.  On  |  ph }  ->  -.  A  e.  |^| { x  e.  On  |  ph } )
62, 5sylbir 213 . . 3  |-  ( ( A  e.  On  /\  ps )  ->  -.  A  e.  |^| { x  e.  On  |  ph }
)
76ex 434 . 2  |-  ( A  e.  On  ->  ( ps  ->  -.  A  e.  |^|
{ x  e.  On  |  ph } ) )
87con2d 115 1  |-  ( A  e.  On  ->  ( A  e.  |^| { x  e.  On  |  ph }  ->  -.  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818    C_ wss 3476   |^|cint 4282   Oncon0 4878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882
This theorem is referenced by:  onminex  6627  oawordeulem  7204  oeeulem  7251  nnawordex  7287  tcrank  8303  alephnbtwn  8453  cardaleph  8471  cardmin  8940  sltval2  29269
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