Theorem onirri 4527

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 4411	. 2 |- Ord A
3		ordirr 4526	. 2 |- ( Ord A -> -. A e. A )
4	2, 3	ax-mp 5	1 |- -. A e. A

Syntax hints:   -. wn 3   e. wcel 2141   Ord word 4347   Oncon0 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-sn 3589  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353

This theorem is referenced by:  ontri2orexmidim  4556  enpr2d  6795  pm54.43  7167  pw1ne1  7206