Theorem ordirr 4663

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 4658. If in the definition of ordinals df-iord 4486, we also required that membership be well-founded on any ordinal (see df-frind 4452), then we could prove ordirr 4663 without ax-setind 4658. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4662	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2203  Ord word 4482

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-v 2814  df-dif 3212  df-sn 3694

This theorem is referenced by:  onirri  4664  nordeq  4665  ordn2lp  4666  orddisj  4667  onprc  4673  nlimsucg  4687  tfr1onlemsucfn  6570  tfr1onlemsucaccv  6571  tfrcllemsucfn  6583  tfrcllemsucaccv  6584  nntr2  6735  1ndom2  7118  unsnfi  7178  nnnninfeq  7418  nninfisol  7423  addnidpig  7650  frecfzennn  10787  hashinfom  11139  hashennn  11141  hashp1i  11173  ennnfonelemg  13146  ctinfom  13171  3dom  16754