Theorem ordirr 4559

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 4554. If in the definition of ordinals df-iord 4384, we also required that membership be well-founded on any ordinal (see df-frind 4350), then we could prove ordirr 4559 without ax-setind 4554. (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 4558	. 2 |- -. A e. A
2	1	a1i 9	1 |- ( Ord A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2160   Ord word 4380

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 2171  ax-setind 4554

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-sn 3613

This theorem is referenced by:  onirri  4560  nordeq  4561  ordn2lp  4562  orddisj  4563  onprc  4569  nlimsucg  4583  tfr1onlemsucfn  6364  tfr1onlemsucaccv  6365  tfrcllemsucfn  6377  tfrcllemsucaccv  6378  nntr2  6527  unsnfi  6946  nnnninfeq  7155  nninfisol  7160  addnidpig  7364  frecfzennn  10456  hashinfom  10789  hashennn  10791  hashp1i  10821  ennnfonelemg  12453  ctinfom  12478