MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onnminsb Structured version   Unicode version

Theorem onnminsb 4776
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 3084 . . . 4  |-  ( A  e.  { x  e.  On  |  ph }  <->  ( A  e.  On  /\  ps ) )
3 ssrab2 3420 . . . . 5  |-  { x  e.  On  |  ph }  C_  On
4 onnmin 4775 . . . . 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 1652    e. wcel 1725   {crab 2701    C_ wss 3312   |^|cint 4042   Oncon0 4573
This theorem is referenced by:  onminex  4779  oawordeulem  6789  oeeulem  6836  nnawordex  6872  tcrank  7800  alephnbtwn  7944  cardaleph  7962  cardmin  8431  sltval2  25603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
  Copyright terms: Public domain W3C validator