Theorem ordirr 4495

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 4490. If in the definition of ordinals df-iord 4321, we also required that membership be well-founded on any ordinal (see df-frind 4287), then we could prove ordirr 4495 without ax-setind 4490. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4494	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2125  Ord word 4317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136  ax-setind 4490

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-v 2711  df-dif 3100  df-sn 3562

This theorem is referenced by:  onirri  4496  nordeq  4497  ordn2lp  4498  orddisj  4499  onprc  4505  nlimsucg  4519  tfr1onlemsucfn  6277  tfr1onlemsucaccv  6278  tfrcllemsucfn  6290  tfrcllemsucaccv  6291  nntr2  6439  unsnfi  6852  addnidpig  7235  frecfzennn  10303  hashinfom  10629  hashennn  10631  hashp1i  10661  ennnfonelemg  12083  ctinfom  12108  nninfalllemn  13520