Theorem ordirr 4542

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 4537. If in the definition of ordinals df-iord 4367, we also required that membership be well-founded on any ordinal (see df-frind 4333), then we could prove ordirr 4542 without ax-setind 4537. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4541	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2148  Ord word 4363

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2740  df-dif 3132  df-sn 3599

This theorem is referenced by:  onirri  4543  nordeq  4544  ordn2lp  4545  orddisj  4546  onprc  4552  nlimsucg  4566  tfr1onlemsucfn  6341  tfr1onlemsucaccv  6342  tfrcllemsucfn  6354  tfrcllemsucaccv  6355  nntr2  6504  unsnfi  6918  nnnninfeq  7126  nninfisol  7131  addnidpig  7335  frecfzennn  10426  hashinfom  10758  hashennn  10760  hashp1i  10790  ennnfonelemg  12404  ctinfom  12429