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Theorem wfelirr 8938
Description: A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 8744. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
wfelirr (𝐴 (𝑅1 “ On) → ¬ 𝐴𝐴)

Proof of Theorem wfelirr
StepHypRef Expression
1 rankon 8908 . . 3 (rank‘𝐴) ∈ On
21onirri 6050 . 2 ¬ (rank‘𝐴) ∈ (rank‘𝐴)
3 rankelb 8937 . 2 (𝐴 (𝑅1 “ On) → (𝐴𝐴 → (rank‘𝐴) ∈ (rank‘𝐴)))
42, 3mtoi 190 1 (𝐴 (𝑅1 “ On) → ¬ 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2157   cuni 4637  cima 5321  Oncon0 5943  cfv 6104  𝑅1cr1 8875  rankcrnk 8876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-om 7299  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-r1 8877  df-rank 8878
This theorem is referenced by:  r1wunlim  9847
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