Theorem onirri 6439

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 6438	. 2 ⊢ Ord 𝐴
3		ordirr 6343	. 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
4	2, 3	ax-mp 5	1 ⊢ ¬ 𝐴 ∈ 𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 2114  Ord word 6324  Oncon0 6325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329

This theorem is referenced by:  onssnel2i  6443  onuninsuci  7792  nlim2  8427  ord1eln01  8433  ord2eln012  8434  oelim2  8533  omopthlem2  8598  harndom  9479  ssttrcl  9636  wfelirr  9749  carduni  9905  pm54.43  9925  alephle  10010  alephfp  10030  pwxpndom2  10588  oldirr  27898  lrrecpo  27949  onsucsuccmpi  36656  onint1  36662  finxpreclem5  37644  wepwsolem  43393