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

Theorem onn0 5753
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 5742 . 2 ∅ ∈ On
21ne0ii 3904 1 On ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wne 2790  c0 3896  Oncon0 5687
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 4754
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 3191  df-dif 3562  df-in 3566  df-ss 3573  df-nul 3897  df-pw 4137  df-uni 4408  df-tr 4718  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690  df-on 5691
This theorem is referenced by:  limon  6990
  Copyright terms: Public domain W3C validator