Theorem onm 4432

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 4423	. . 3 |- (/) e. On
2		0ex 4156	. . . 4 |- (/) e. _V
3		eleq1 2256	. . . 4 |- ( x = (/) -> ( x e. On <-> (/) e. On ) )
4	2, 3	ceqsexv 2799	. . 3 |- ( E. x ( x = (/) /\ x e. On ) <-> (/) e. On )
5	1, 4	mpbir 146	. 2 |- E. x ( x = (/) /\ x e. On )
6		exsimpr 1629	. 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 1364   E.wex 1503   e. wcel 2164   (/)c0 3446   Oncon0 4394

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-nul 4155

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399

This theorem is referenced by: (None)