Theorem ordirr 4631

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 4626. If in the definition of ordinals df-iord 4454, we also required that membership be well-founded on any ordinal (see df-frind 4420), then we could prove ordirr 4631 without ax-setind 4626. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4630	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2200  Ord word 4450

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 4626

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  4632  nordeq  4633  ordn2lp  4634  orddisj  4635  onprc  4641  nlimsucg  4655  tfr1onlemsucfn  6476  tfr1onlemsucaccv  6477  tfrcllemsucfn  6489  tfrcllemsucaccv  6490  nntr2  6639  1ndom2  7014  unsnfi  7069  nnnninfeq  7283  nninfisol  7288  addnidpig  7511  frecfzennn  10635  hashinfom  10987  hashennn  10989  hashp1i  11019  ennnfonelemg  12960  ctinfom  12985