Theorem ordirr 4225

Description: Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4224	. 2 ⊢ ¬ A ∈ A
2	1	a1i 9	1 ⊢ (Ord A → ¬ A ∈ A)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1390  Ord word 4065

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-sn 3373

This theorem is referenced by:  onprc  4230  nlimsucg  4242  addnidpig  6320  frecfzennn  8884