Theorem ordirr 4574

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 4569. If in the definition of ordinals df-iord 4397, we also required that membership be well-founded on any ordinal (see df-frind 4363), then we could prove ordirr 4574 without ax-setind 4569. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4573	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2164  Ord word 4393

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-v 2762  df-dif 3155  df-sn 3624

This theorem is referenced by:  onirri  4575  nordeq  4576  ordn2lp  4577  orddisj  4578  onprc  4584  nlimsucg  4598  tfr1onlemsucfn  6393  tfr1onlemsucaccv  6394  tfrcllemsucfn  6406  tfrcllemsucaccv  6407  nntr2  6556  unsnfi  6975  nnnninfeq  7187  nninfisol  7192  addnidpig  7396  frecfzennn  10497  hashinfom  10849  hashennn  10851  hashp1i  10881  ennnfonelemg  12560  ctinfom  12585