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Theorem inton 5999
 Description: The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
Assertion
Ref Expression
inton On = ∅

Proof of Theorem inton
StepHypRef Expression
1 0elon 5995 . 2 ∅ ∈ On
2 int0el 4699 . 2 (∅ ∈ On → On = ∅)
31, 2ax-mp 5 1 On = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1653   ∈ wcel 2157  ∅c0 4116  ∩ cint 4668  Oncon0 5942 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-nul 4984 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-in 3777  df-ss 3784  df-nul 4117  df-pw 4352  df-uni 4630  df-int 4669  df-tr 4947  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-ord 5945  df-on 5946 This theorem is referenced by: (None)
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