Theorem onirri 4554

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 4438	. 2 |- Ord A
3		ordirr 4553	. 2 |- ( Ord A -> -. A e. A )
4	2, 3	ax-mp 5	1 |- -. A e. A

Syntax hints:   -. wn 3   e. wcel 2158   Ord word 4374   Oncon0 4375

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-setind 4548

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-in 3147  df-ss 3154  df-sn 3610  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380

This theorem is referenced by:  ontri2orexmidim  4583  enpr2d  6831  pm54.43  7203  pw1ne1  7242