Theorem nlim0 4149

Description: The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
nlim0	⊢ ¬ Lim ∅

Proof of Theorem nlim0

Step	Hyp	Ref	Expression
1		noel 3255	. . 3 ⊢ ¬ ∅ ∈ ∅
2		simp2 939	. . 3 ⊢ ((Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅) → ∅ ∈ ∅)
3	1, 2	mto 620	. 2 ⊢ ¬ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅)
4		dflim2 4125	. 2 ⊢ (Lim ∅ ↔ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅))
5	3, 4	mtbir 628	1 ⊢ ¬ Lim ∅

Syntax hints:  ¬ wn 3   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∅c0 3251  ∪ cuni 3601  Ord word 4117  Lim wlim 4119

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-nul 3252  df-ilim 4124

This theorem is referenced by: (None)