Theorem ordirr 4634

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 4629. If in the definition of ordinals df-iord 4457, we also required that membership be well-founded on any ordinal (see df-frind 4423), then we could prove ordirr 4634 without ax-setind 4629. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4633	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2200  Ord word 4453

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-v 2801  df-dif 3199  df-sn 3672

This theorem is referenced by:  onirri  4635  nordeq  4636  ordn2lp  4637  orddisj  4638  onprc  4644  nlimsucg  4658  tfr1onlemsucfn  6492  tfr1onlemsucaccv  6493  tfrcllemsucfn  6505  tfrcllemsucaccv  6506  nntr2  6657  1ndom2  7034  unsnfi  7089  nnnninfeq  7303  nninfisol  7308  addnidpig  7531  frecfzennn  10656  hashinfom  11008  hashennn  11010  hashp1i  11040  ennnfonelemg  12982  ctinfom  13007  3dom  16381