Theorem ontrci 4518

Description: An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)

Hypothesis

Ref	Expression
on.1	|- A e. On

Assertion

Ref	Expression
ontrci	|- Tr A

Proof of Theorem ontrci

Step	Hyp	Ref	Expression
1		on.1	. . 3 |- A e. On
2	1	onordi 4517	. 2 |- Ord A
3		ordtr 4469	. 2 |- ( Ord A -> Tr A )
4	2, 3	ax-mp 5	1 |- Tr A

Syntax hints:   e. wcel 2200   Tr wtr 4182   Ord word 4453   Oncon0 4454

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-uni 3889  df-tr 4183  df-iord 4457  df-on 4459

This theorem is referenced by:  onunisuci  4523  exmidonfinlem  7371  bj-el2oss1o  16138  nnsf  16371