Theorem nlim0 3022

Description: The empty set is not a limit ordinal.

Assertion

Ref	Expression
nlim0	|- -. Lim (/)

Proof of Theorem nlim0

Step	Hyp	Ref	Expression
1		eqid 1473	. 2 |- (/) = (/)
2		df-lim 2948	. . . 4 |- (Lim (/) <-> (Ord (/) /\ (/) =/= (/) /\ (/) = U.(/)))
3		3simp2 788	. . . 4 |- ((Ord (/) /\ (/) =/= (/) /\ (/) = U.(/)) -> (/) =/= (/))
4	2, 3	sylbi 199	. . 3 |- (Lim (/) -> (/) =/= (/))
5	4	necon2bi 1609	. 2 |- ((/) = (/) -> -. Lim (/))
6	1, 5	ax-mp 7	1 |- -. Lim (/)

Syntax hints:  -. wn 2   /\ w3a 774   = wceq 954   =/= wne 1582  (/)c0 2276  U.cuni 2498  Ord word 2942  Lim wlim 2944

This theorem is referenced by:  0ellim 3026  dflim3 3113  tz7.44lem1 3918  dfrdg2 3924

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 776  df-cleq 1467  df-ne 1584  df-lim 2948

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