Theorem ordirr 4689

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 4684. If in the definition of ordinals df-iord 4511, we also required that membership be well-founded on any ordinal (see df-frind 4477), then we could prove ordirr 4689 without ax-setind 4684. (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 4688	. 2 |- -. A e. A
2	1	a1i 9	1 |- ( Ord A -> -. A e. A )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   e. wcel 2209   Ord word 4507

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-v 2823  df-dif 3222  df-sn 3715

This theorem is used by:  onirri  4690  nordeq  4691  ordn2lp  4692  orddisj  4693  onprc  4699  nlimsucg  4713  tfr1onlemsucfn  6611  tfr1onlemsucaccv  6612  tfrcllemsucfn  6624  tfrcllemsucaccv  6625  nntr2  6776  1ndom2  7166  unsnfi  7226  nnnninfeq  7468  nninfisol  7473  addnidpig  7703  frecfzennn  10863  hashinfom  11217  hashennn  11219  hashp1i  11251  ennnfonelemg  13294  ctinfom  13319  3dom  17018