Theorem ordeirr 2217

Description: Epsilon irreflexivity of ordinals: no ordinal is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. For ordinals, we can prove this without invoking the Axiom of Regularity.

Assertion

Ref	Expression
ordeirr	⊢ (Ord A → ¬ A ∈ A)

Proof of Theorem ordeirr

Step	Hyp	Ref	Expression
1		ordfr 2214	. 2 ⊢ (Ord A → E Fr A)
2		efrirr 2180	. 2 ⊢ (E Fr A → ¬ A ∈ A)
3	1, 2	syl 12	1 ⊢ (Ord A → ¬ A ∈ A)

Syntax hints:  ¬ wn 1   → wi 2   ∈ wcel 1092  Ecep 2056   Fr wfr 2061  Ord word 2198

This theorem is referenced by:  nordeq 2218  ordn2lp 2219  ordtri3or 2230  ordtri1 2231  ordtri3 2234  orddisj 2236  onprc 2240  ordunidif 2260  ordnbtwn 2316  oneirr 2345  onssneli 2349  nlimsuc 2363  nnlim 2385  limom 2387  tfrlem10 2958  tfrlem13 2961  limensuci 3401  infensuc 3484  ondomcard 3663  addnidpi 3822

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-eprel 2122  df-fr 2169  df-we 2186  df-ord 2202

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