Theorem onm 4156

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 4147	. . 3 |- (/) e. On
2		0ex 3905	. . . 4 |- (/) e. _V
3		eleq1 2141	. . . 4 |- ( x = (/) -> ( x e. On <-> (/) e. On ) )
4	2, 3	ceqsexv 2638	. . 3 |- ( E. x ( x = (/) /\ x e. On ) <-> (/) e. On )
5	1, 4	mpbir 144	. 2 |- E. x ( x = (/) /\ x e. On )
6		exsimpr 1549	. 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 102   = wceq 1284   E.wex 1421   e. wcel 1433   (/)c0 3251   Oncon0 4118

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-nul 3904

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123

This theorem is referenced by: (None)