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

Theorem onn0 4472
Description: The class of all ordinal numbers in not empty. (Contributed by NM, 17-Sep-1995.)
Assertion
Ref Expression
onn0  |-  On  =/=  (/)

Proof of Theorem onn0
StepHypRef Expression
1 0elon 4461 . 2  |-  (/)  e.  On
2 ne0i 3474 . 2  |-  ( (/)  e.  On  ->  On  =/=  (/) )
31, 2ax-mp 8 1  |-  On  =/=  (/)
Colors of variables: wff set class
Syntax hints:    e. wcel 1696    =/= wne 2459   (/)c0 3468   Oncon0 4408
This theorem is referenced by:  limon  4643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-uni 3844  df-tr 4130  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
  Copyright terms: Public domain W3C validator