Theorem ordirr 4608

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 4603. If in the definition of ordinals df-iord 4431, we also required that membership be well-founded on any ordinal (see df-frind 4397), then we could prove ordirr 4608 without ax-setind 4603. (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 4607	. 2 |- -. A e. A
2	1	a1i 9	1 |- ( Ord A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2178   Ord word 4427

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-v 2778  df-dif 3176  df-sn 3649

This theorem is referenced by:  onirri  4609  nordeq  4610  ordn2lp  4611  orddisj  4612  onprc  4618  nlimsucg  4632  tfr1onlemsucfn  6449  tfr1onlemsucaccv  6450  tfrcllemsucfn  6462  tfrcllemsucaccv  6463  nntr2  6612  1ndom2  6987  unsnfi  7042  nnnninfeq  7256  nninfisol  7261  addnidpig  7484  frecfzennn  10608  hashinfom  10960  hashennn  10962  hashp1i  10992  ennnfonelemg  12889  ctinfom  12914