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Theorem onm 4402
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 4393 . . 3  |-  (/)  e.  On
2 0ex 4131 . . . 4  |-  (/)  e.  _V
3 eleq1 2240 . . . 4  |-  ( x  =  (/)  ->  ( x  e.  On  <->  (/)  e.  On ) )
42, 3ceqsexv 2777 . . 3  |-  ( E. x ( x  =  (/)  /\  x  e.  On ) 
<->  (/)  e.  On )
51, 4mpbir 146 . 2  |-  E. x
( x  =  (/)  /\  x  e.  On )
6 exsimpr 1618 . 2  |-  ( E. x ( x  =  (/)  /\  x  e.  On )  ->  E. x  x  e.  On )
75, 6ax-mp 5 1  |-  E. x  x  e.  On
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353   E.wex 1492    e. wcel 2148   (/)c0 3423   Oncon0 4364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-nul 4130
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-uni 3811  df-tr 4103  df-iord 4367  df-on 4369
This theorem is referenced by: (None)
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