Theorem onirri 4641

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 4523	. 2 |- Ord A
3		ordirr 4640	. 2 |- ( Ord A -> -. A e. A )
4	2, 3	ax-mp 5	1 |- -. A e. A

Syntax hints:   -. wn 3   e. wcel 2202   Ord word 4459   Oncon0 4460

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-sn 3675  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465

This theorem is referenced by:  ontri2orexmidim  4670  enpr2d  6996  pm54.43  7394  pw1ne1  7446