Theorem ordirr 4541

Description: Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. The present proof requires ax-setind 4536. If in the definition of ordinals df-iord 4366, we also required that membership be well-founded on any ordinal (see df-frind 4332), then we could prove ordirr 4541 without ax-setind 4536. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4540	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2148  Ord word 4362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4536

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2739  df-dif 3131  df-sn 3598

This theorem is referenced by:  onirri  4542  nordeq  4543  ordn2lp  4544  orddisj  4545  onprc  4551  nlimsucg  4565  tfr1onlemsucfn  6340  tfr1onlemsucaccv  6341  tfrcllemsucfn  6353  tfrcllemsucaccv  6354  nntr2  6503  unsnfi  6917  nnnninfeq  7125  nninfisol  7130  addnidpig  7334  frecfzennn  10425  hashinfom  10757  hashennn  10759  hashp1i  10789  ennnfonelemg  12403  ctinfom  12428