Theorem onm 4323

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 4314	. . 3 |- (/) e. On
2		0ex 4055	. . . 4 |- (/) e. _V
3		eleq1 2202	. . . 4 |- ( x = (/) -> ( x e. On <-> (/) e. On ) )
4	2, 3	ceqsexv 2725	. . 3 |- ( E. x ( x = (/) /\ x e. On ) <-> (/) e. On )
5	1, 4	mpbir 145	. 2 |- E. x ( x = (/) /\ x e. On )
6		exsimpr 1597	. 2 |- ( E. x ( x = (/) /\ x e. On ) -> E. x x e. On )
7	5, 6	ax-mp 5	1 |- E. x x e. On

Syntax hints:   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   (/)c0 3363   Oncon0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290

This theorem is referenced by: (None)