Theorem ordirr 4293

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 4288. If in the definition of ordinals df-iord 4129, we also required that membership be well-founded on any ordinal (see df-frind 4095), then we could prove ordirr 4293 without ax-setind 4288. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4292	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1434  Ord word 4125

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-setind 4288

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-v 2604  df-dif 2976  df-sn 3412

This theorem is referenced by:  onirri  4294  nordeq  4295  ordn2lp  4296  orddisj  4297  onprc  4303  nlimsucg  4317  tfr1onlemsucfn  5989  tfr1onlemsucaccv  5990  tfrcllemsucfn  6002  tfrcllemsucaccv  6003  unsnfi  6439  addnidpig  6588  frecfzennn  9508  sizeinf  9802  sizeennn  9804  sizep1i  9834