Theorem ordirr 4687

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 4682. If in the definition of ordinals df-iord 4509, we also required that membership be well-founded on any ordinal (see df-frind 4475), then we could prove ordirr 4687 without ax-setind 4682. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4686	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2209  Ord word 4505

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 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 4682

This theorem 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 3714

This theorem is referenced by:  onirri  4688  nordeq  4689  ordn2lp  4690  orddisj  4691  onprc  4697  nlimsucg  4711  tfr1onlemsucfn  6605  tfr1onlemsucaccv  6606  tfrcllemsucfn  6618  tfrcllemsucaccv  6619  nntr2  6770  1ndom2  7160  unsnfi  7220  nnnninfeq  7462  nninfisol  7467  addnidpig  7697  frecfzennn  10846  hashinfom  11200  hashennn  11202  hashp1i  11234  ennnfonelemg  13277  ctinfom  13302  3dom  17001