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Theorem onnminsb 6636
 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 3226 . . . 4
3 ssrab2 3543 . . . . 5
4 onnmin 6635 . . . . 5
53, 4mpan 674 . . . 4
62, 5sylbir 216 . . 3
76ex 435 . 2
87con2d 118 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wa 370   wceq 1437   wcel 1867  crab 2777   wss 3433  cint 4249  con0 5433 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652  ax-un 6588 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-br 4418  df-opab 4476  df-tr 4512  df-eprel 4756  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-ord 5436  df-on 5437 This theorem is referenced by:  onminex  6639  oawordeulem  7254  oeeulem  7301  nnawordex  7337  tcrank  8345  alephnbtwn  8491  cardaleph  8509  cardmin  8978  sltval2  30371
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