Theorem nlim0 4311

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 3362	. . 3 |- -. (/) e. (/)
2		simp2 982	. . 3 |- ( ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) ) -> (/) e. (/) )
3	1, 2	mto 651	. 2 |- -. ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) )
4		dflim2 4287	. 2 |- ( Lim (/) <-> ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) ) )
5	3, 4	mtbir 660	1 |- -. Lim (/)

Syntax hints:   -. wn 3   /\ w3a 962   = wceq 1331   e. wcel 1480   (/)c0 3358   U.cuni 3731   Ord word 4279   Lim wlim 4281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-nul 3359  df-ilim 4286

This theorem is referenced by: (None)