Theorem ordirr 4519

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 4514. If in the definition of ordinals df-iord 4344, we also required that membership be well-founded on any ordinal (see df-frind 4310), then we could prove ordirr 4519 without ax-setind 4514. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4518	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2136  Ord word 4340

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-v 2728  df-dif 3118  df-sn 3582

This theorem is referenced by:  onirri  4520  nordeq  4521  ordn2lp  4522  orddisj  4523  onprc  4529  nlimsucg  4543  tfr1onlemsucfn  6308  tfr1onlemsucaccv  6309  tfrcllemsucfn  6321  tfrcllemsucaccv  6322  nntr2  6471  unsnfi  6884  nnnninfeq  7092  nninfisol  7097  addnidpig  7277  frecfzennn  10361  hashinfom  10691  hashennn  10693  hashp1i  10723  ennnfonelemg  12336  ctinfom  12361