Theorem onirri 4544

Description: An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)

Hypothesis

Ref	Expression
onirri.1	|- A e. On

Assertion

Ref	Expression
onirri	|- -. A e. A

Proof of Theorem onirri

Step	Hyp	Ref	Expression
1		onirri.1	. . 3 |- A e. On
2	1	onordi 4428	. 2 |- Ord A
3		ordirr 4543	. 2 |- ( Ord A -> -. A e. A )
4	2, 3	ax-mp 5	1 |- -. A e. A

Syntax hints:   -. wn 3   e. wcel 2148   Ord word 4364   Oncon0 4365

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-sn 3600  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370

This theorem is referenced by:  ontri2orexmidim  4573  enpr2d  6819  pm54.43  7191  pw1ne1  7230