Theorem ordirr 4646

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 4641. If in the definition of ordinals df-iord 4469, we also required that membership be well-founded on any ordinal (see df-frind 4435), then we could prove ordirr 4646 without ax-setind 4641. (Contributed by NM, 2-Jan-1994.)

Assertion

Ref	Expression
ordirr	⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4645	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2202  Ord word 4465

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 2213  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-v 2805  df-dif 3203  df-sn 3679

This theorem is referenced by:  onirri  4647  nordeq  4648  ordn2lp  4649  orddisj  4650  onprc  4656  nlimsucg  4670  tfr1onlemsucfn  6549  tfr1onlemsucaccv  6550  tfrcllemsucfn  6562  tfrcllemsucaccv  6563  nntr2  6714  1ndom2  7094  unsnfi  7154  nnnninfeq  7370  nninfisol  7375  addnidpig  7599  frecfzennn  10732  hashinfom  11084  hashennn  11086  hashp1i  11118  ennnfonelemg  13085  ctinfom  13110  3dom  16688