Theorem ontrci 4572

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 4571	. 2 |- Ord A
3		ordtr 4523	. 2 |- ( Ord A -> Tr A )
4	2, 3	ax-mp 5	1 |- Tr A

This proof depends on syntax axioms:   e. wcel 2209   Tr wtr 4229   Ord word 4507   Oncon0 4508

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-uni 3936  df-tr 4230  df-iord 4511  df-on 4513

This theorem is used by:  onunisuci  4577  exmidonfinlem  7545  bj-el2oss1o  16802  nnsf  17048