Theorem onm 4500

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 4491	. . 3 |- (/) e. On
2		0ex 4217	. . . 4 |- (/) e. _V
3		eleq1 2293	. . . 4 |- ( x = (/) -> ( x e. On <-> (/) e. On ) )
4	2, 3	ceqsexv 2841	. . 3 |- ( E. x ( x = (/) /\ x e. On ) <-> (/) e. On )
5	1, 4	mpbir 146	. 2 |- E. x ( x = (/) /\ x e. On )
6		exsimpr 1666	. 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 104   = wceq 1397   E.wex 1540   e. wcel 2201   (/)c0 3493   Oncon0 4462

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212  ax-nul 4216

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-dif 3201  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-uni 3895  df-tr 4189  df-iord 4465  df-on 4467

This theorem is referenced by: (None)