Theorem ordirr 4640

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 4635. If in the definition of ordinals df-iord 4463, we also required that membership be well-founded on any ordinal (see df-frind 4429), then we could prove ordirr 4640 without ax-setind 4635. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4639	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2202  Ord word 4459

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-v 2804  df-dif 3202  df-sn 3675

This theorem is referenced by:  onirri  4641  nordeq  4642  ordn2lp  4643  orddisj  4644  onprc  4650  nlimsucg  4664  tfr1onlemsucfn  6506  tfr1onlemsucaccv  6507  tfrcllemsucfn  6519  tfrcllemsucaccv  6520  nntr2  6671  1ndom2  7051  unsnfi  7111  nnnninfeq  7327  nninfisol  7332  addnidpig  7556  frecfzennn  10689  hashinfom  11041  hashennn  11043  hashp1i  11075  ennnfonelemg  13029  ctinfom  13054  3dom  16613