Theorem ordirr 4417

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 4412. If in the definition of ordinals df-iord 4248, we also required that membership be well-founded on any ordinal (see df-frind 4214), then we could prove ordirr 4417 without ax-setind 4412. (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 4416	. 2 |- -. A e. A
2	1	a1i 9	1 |- ( Ord A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1463   Ord word 4244

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-setind 4412

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-v 2659  df-dif 3039  df-sn 3499

This theorem is referenced by:  onirri  4418  nordeq  4419  ordn2lp  4420  orddisj  4421  onprc  4427  nlimsucg  4441  tfr1onlemsucfn  6191  tfr1onlemsucaccv  6192  tfrcllemsucfn  6204  tfrcllemsucaccv  6205  nntr2  6353  unsnfi  6760  addnidpig  7092  frecfzennn  10092  hashinfom  10417  hashennn  10419  hashp1i  10449  ennnfonelemg  11761  ctinfom  11786  nninfalllemn  12892