Theorem onm 4138

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

Step	Hyp	Ref	Expression
1		0elon 4129	. . 3 |- (/) e. On
2		0ex 3884	. . . 4 |- (/) e. _V
3		eleq1 2100	. . . 4 |- ( x = (/) -> ( x e. On <-> (/) e. On ) )
4	2, 3	ceqsexv 2593	. . 3 |- ( E. x ( x = (/) /\ x e. On ) <-> (/) e. On )
5	1, 4	mpbir 134	. 2 |- E. x ( x = (/) /\ x e. On )
6		exsimpr 1509	. 2 |- ( E. x ( x = (/) /\ x e. On ) -> E. x x e. On )
7	5, 6	ax-mp 7	1 |- E. x x e. On

Syntax hints:   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   (/)c0 3224   Oncon0 4100

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105

This theorem is referenced by: (None)