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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. is the wff resulting from the substitution of for in wff . (Contributed by NM, 9-Nov-2003.)
Hypothesis
Ref Expression
onnminsb.1
Assertion
Ref Expression
onnminsb
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem onnminsb
StepHypRef Expression
1 onnminsb.1 . . . . 5
21elrab 3084 . . . 4
3 ssrab2 3420 . . . . 5
4 onnmin 4775 . . . . 5
53, 4mpan 652 . . . 4
62, 5sylbir 205 . . 3
76ex 424 . 2
87con2d 109 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  crab 2701   wss 3312  cint 4042  con0 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
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