MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onn0 Structured version   Visualization version   GIF version

Theorem onn0 6429
Description: The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
Assertion
Ref Expression
onn0 On ≠ ∅

Proof of Theorem onn0
StepHypRef Expression
1 0elon 6418 . 2 ∅ ∈ On
21ne0ii 4337 1 On ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wne 2940  c0 4322  Oncon0 6364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-tr 5266  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368
This theorem is referenced by:  limon  7823
  Copyright terms: Public domain W3C validator