Theorem ordirr 4669

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 4664. If in the definition of ordinals df-iord 4492, we also required that membership be well-founded on any ordinal (see df-frind 4458), then we could prove ordirr 4669 without ax-setind 4664. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4668	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2205  Ord word 4488

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 2216  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-v 2817  df-dif 3216  df-sn 3700

This theorem is referenced by:  onirri  4670  nordeq  4671  ordn2lp  4672  orddisj  4673  onprc  4679  nlimsucg  4693  tfr1onlemsucfn  6584  tfr1onlemsucaccv  6585  tfrcllemsucfn  6597  tfrcllemsucaccv  6598  nntr2  6749  1ndom2  7132  unsnfi  7192  nnnninfeq  7432  nninfisol  7437  addnidpig  7667  frecfzennn  10812  hashinfom  11166  hashennn  11168  hashp1i  11200  ennnfonelemg  13238  ctinfom  13263  3dom  16874