Theorem onirri 6431

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
on.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
onirri	⊢ ¬ 𝐴 ∈ 𝐴

Proof of Theorem onirri

Step	Hyp	Ref	Expression
1		on.1	. . 3 ⊢ 𝐴 ∈ On
2	1	onordi 6430	. 2 ⊢ Ord 𝐴
3		ordirr 6335	. 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
4	2, 3	ax-mp 5	1 ⊢ ¬ 𝐴 ∈ 𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 2113  Ord word 6316  Oncon0 6317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321

This theorem is referenced by:  onssnel2i  6435  onuninsuci  7782  nlim2  8417  ord1eln01  8423  ord2eln012  8424  oelim2  8523  omopthlem2  8588  harndom  9467  ssttrcl  9624  wfelirr  9737  carduni  9893  pm54.43  9913  alephle  9998  alephfp  10018  pwxpndom2  10576  oldirr  27886  lrrecpo  27937  onsucsuccmpi  36637  onint1  36643  finxpreclem5  37596  wepwsolem  43280