Theorem ordirr 4572

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 4567. If in the definition of ordinals df-iord 4395, we also required that membership be well-founded on any ordinal (see df-frind 4361), then we could prove ordirr 4572 without ax-setind 4567. (Contributed by NM, 2-Jan-1994.)

Assertion

Ref	Expression
ordirr	|- ( Ord A -> -. A e. A )

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4571	. 2 |- -. A e. A
2	1	a1i 9	1 |- ( Ord A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2164   Ord word 4391

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 4567

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  4573  nordeq  4574  ordn2lp  4575  orddisj  4576  onprc  4582  nlimsucg  4596  tfr1onlemsucfn  6388  tfr1onlemsucaccv  6389  tfrcllemsucfn  6401  tfrcllemsucaccv  6402  nntr2  6551  unsnfi  6970  nnnninfeq  7181  nninfisol  7186  addnidpig  7390  frecfzennn  10491  hashinfom  10843  hashennn  10845  hashp1i  10875  ennnfonelemg  12554  ctinfom  12579