Theorem onirri 4591

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 4473	. 2 |- Ord A
3		ordirr 4590	. 2 |- ( Ord A -> -. A e. A )
4	2, 3	ax-mp 5	1 |- -. A e. A

Syntax hints:   -. wn 3   e. wcel 2176   Ord word 4409   Oncon0 4410

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-sn 3639  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415

This theorem is referenced by:  ontri2orexmidim  4620  enpr2d  6911  pm54.43  7298  pw1ne1  7341