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Theorem inton 5741
 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 5737 . 2 ∅ ∈ On
2 int0el 4473 . 2 (∅ ∈ On → On = ∅)
31, 2ax-mp 5 1 On = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480   ∈ wcel 1987  ∅c0 3891  ∩ cint 4440  Oncon0 5682 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-pw 4132  df-uni 4403  df-int 4441  df-tr 4713  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686 This theorem is referenced by: (None)
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