Theorem onirri 4286

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 ∈ On

Assertion

Ref	Expression
onirri	⊢ ¬ 𝐴 ∈ 𝐴

Proof of Theorem onirri

Step	Hyp	Ref	Expression
1		onirri.1	. . 3 ⊢ 𝐴 ∈ On
2	1	onordi 4181	. 2 ⊢ Ord 𝐴
3		ordirr 4285	. 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
4	2, 3	ax-mp 7	1 ⊢ ¬ 𝐴 ∈ 𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 1433  Ord word 4117  Oncon0 4118

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-sn 3404  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123

This theorem is referenced by:  pm54.43  6459