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Theorem onm 4228
Description: The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)
Assertion
Ref Expression
onm  |-  E. x  x  e.  On

Proof of Theorem onm
StepHypRef Expression
1 0elon 4219 . . 3  |-  (/)  e.  On
2 0ex 3966 . . . 4  |-  (/)  e.  _V
3 eleq1 2150 . . . 4  |-  ( x  =  (/)  ->  ( x  e.  On  <->  (/)  e.  On ) )
42, 3ceqsexv 2658 . . 3  |-  ( E. x ( x  =  (/)  /\  x  e.  On ) 
<->  (/)  e.  On )
51, 4mpbir 144 . 2  |-  E. x
( x  =  (/)  /\  x  e.  On )
6 exsimpr 1554 . 2  |-  ( E. x ( x  =  (/)  /\  x  e.  On )  ->  E. x  x  e.  On )
75, 6ax-mp 7 1  |-  E. x  x  e.  On
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   (/)c0 3286   Oncon0 4190
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3965
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195
This theorem is referenced by: (None)
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