Theorem ordirr 4691

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 4686. If in the definition of ordinals df-iord 4513, we also required that membership be well-founded on any ordinal (see df-frind 4479), then we could prove ordirr 4691 without ax-setind 4686. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4690	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2209  Ord word 4509

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-setind 4686

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-v 2823  df-dif 3222  df-sn 3715

This theorem is used by:  onirri  4692  nordeq  4693  ordn2lp  4694  orddisj  4695  onprc  4701  nlimsucg  4715  tfr1onlemsucfn  6615  tfr1onlemsucaccv  6616  tfrcllemsucfn  6628  tfrcllemsucaccv  6629  nntr2  6780  1ndom2  7170  unsnfi  7230  nnnninfeq  7474  nninfisol  7479  addnidpig  7709  frecfzennn  10884  hashinfom  11239  hashennn  11241  hashp1i  11273  ennnfonelemg  13352  ctinfom  13377  3dom  17223