Theorem nlim0 4212

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 3288	. . 3 |- -. (/) e. (/)
2		simp2 944	. . 3 |- ( ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) ) -> (/) e. (/) )
3	1, 2	mto 623	. 2 |- -. ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) )
4		dflim2 4188	. 2 |- ( Lim (/) <-> ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) ) )
5	3, 4	mtbir 631	1 |- -. Lim (/)

Syntax hints:   -. wn 3   /\ w3a 924   = wceq 1289   e. wcel 1438   (/)c0 3284   U.cuni 3648   Ord word 4180   Lim wlim 4182

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-nul 3285  df-ilim 4187

This theorem is referenced by: (None)