Theorem ordirr 4526

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 4521. If in the definition of ordinals df-iord 4351, we also required that membership be well-founded on any ordinal (see df-frind 4317), then we could prove ordirr 4526 without ax-setind 4521. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4525	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2141  Ord word 4347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-sn 3589

This theorem is referenced by:  onirri  4527  nordeq  4528  ordn2lp  4529  orddisj  4530  onprc  4536  nlimsucg  4550  tfr1onlemsucfn  6319  tfr1onlemsucaccv  6320  tfrcllemsucfn  6332  tfrcllemsucaccv  6333  nntr2  6482  unsnfi  6896  nnnninfeq  7104  nninfisol  7109  addnidpig  7298  frecfzennn  10382  hashinfom  10712  hashennn  10714  hashp1i  10745  ennnfonelemg  12358  ctinfom  12383