Theorem ordirr 4540

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 4535. If in the definition of ordinals df-iord 4365, we also required that membership be well-founded on any ordinal (see df-frind 4331), then we could prove ordirr 4540 without ax-setind 4535. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4539	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2148  Ord word 4361

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 4535

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  4541  nordeq  4542  ordn2lp  4543  orddisj  4544  onprc  4550  nlimsucg  4564  tfr1onlemsucfn  6338  tfr1onlemsucaccv  6339  tfrcllemsucfn  6351  tfrcllemsucaccv  6352  nntr2  6501  unsnfi  6915  nnnninfeq  7123  nninfisol  7128  addnidpig  7332  frecfzennn  10421  hashinfom  10751  hashennn  10753  hashp1i  10783  ennnfonelemg  12396  ctinfom  12421