Theorem ordirr 4638

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 4633. If in the definition of ordinals df-iord 4461, we also required that membership be well-founded on any ordinal (see df-frind 4427), then we could prove ordirr 4638 without ax-setind 4633. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4637	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2200  Ord word 4457

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-v 2802  df-dif 3200  df-sn 3673

This theorem is referenced by:  onirri  4639  nordeq  4640  ordn2lp  4641  orddisj  4642  onprc  4648  nlimsucg  4662  tfr1onlemsucfn  6501  tfr1onlemsucaccv  6502  tfrcllemsucfn  6514  tfrcllemsucaccv  6515  nntr2  6666  1ndom2  7046  unsnfi  7106  nnnninfeq  7321  nninfisol  7326  addnidpig  7549  frecfzennn  10681  hashinfom  11033  hashennn  11035  hashp1i  11067  ennnfonelemg  13017  ctinfom  13042  3dom  16537