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Theorem onn0 4157
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 4149 . 2 ∅ ∈ On
2 ne0i 3258 . 2 (∅ ∈ On → On ≠ ∅)
31, 2ax-mp 7 1 On ≠ ∅
Colors of variables: wff set class
Syntax hints:  wcel 1434  wne 2246  c0 3252  Oncon0 4120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-nul 3906
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-uni 3604  df-tr 3878  df-iord 4123  df-on 4125
This theorem is referenced by: (None)
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