Theorem ordirr 4635

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 4630. If in the definition of ordinals df-iord 4458, we also required that membership be well-founded on any ordinal (see df-frind 4424), then we could prove ordirr 4635 without ax-setind 4630. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4634	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2200  Ord word 4454

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 4630

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  4636  nordeq  4637  ordn2lp  4638  orddisj  4639  onprc  4645  nlimsucg  4659  tfr1onlemsucfn  6497  tfr1onlemsucaccv  6498  tfrcllemsucfn  6510  tfrcllemsucaccv  6511  nntr2  6662  1ndom2  7039  unsnfi  7097  nnnninfeq  7311  nninfisol  7316  addnidpig  7539  frecfzennn  10665  hashinfom  11017  hashennn  11019  hashp1i  11050  ennnfonelemg  12995  ctinfom  13020  3dom  16465