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Theorem onn0 4103
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 4095 . 2 On
2 ne0i 3224 . 2 (∅ On → On ≠ ∅)
31, 2ax-mp 7 1 On ≠ ∅
Colors of variables: wff set class
Syntax hints:   wcel 1390  wne 2201  c0 3218  Oncon0 4066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071
This theorem is referenced by: (None)
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