Theorem ordirr 4666

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 4661. If in the definition of ordinals df-iord 4489, we also required that membership be well-founded on any ordinal (see df-frind 4455), then we could prove ordirr 4666 without ax-setind 4661. (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 4665	. 2 |- -. A e. A
2	1	a1i 9	1 |- ( Ord A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2205   Ord word 4485

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-v 2817  df-dif 3215  df-sn 3697

This theorem is referenced by:  onirri  4667  nordeq  4668  ordn2lp  4669  orddisj  4670  onprc  4676  nlimsucg  4690  tfr1onlemsucfn  6573  tfr1onlemsucaccv  6574  tfrcllemsucfn  6586  tfrcllemsucaccv  6587  nntr2  6738  1ndom2  7121  unsnfi  7181  nnnninfeq  7421  nninfisol  7426  addnidpig  7653  frecfzennn  10792  hashinfom  11145  hashennn  11147  hashp1i  11179  ennnfonelemg  13171  ctinfom  13196  3dom  16779