Theorem nlim0 3033

Description: The empty set is not a limit ordinal.

Assertion

Ref	Expression
nlim0	|- -. Lim (/)

Proof of Theorem nlim0

Step	Hyp	Ref	Expression
1		eqid 1478	. 2 |- (/) = (/)
2		df-lim 2959	. . . 4 |- (Lim (/) <-> (Ord (/) /\ (/) =/= (/) /\ (/) = U.(/)))
3		3simp2 791	. . . 4 |- ((Ord (/) /\ (/) =/= (/) /\ (/) = U.(/)) -> (/) =/= (/))
4	2, 3	sylbi 199	. . 3 |- (Lim (/) -> (/) =/= (/))
5	4	necon2bi 1615	. 2 |- ((/) = (/) -> -. Lim (/))
6	1, 5	ax-mp 7	1 |- -. Lim (/)

Syntax hints:  -. wn 2   /\ w3a 777   = wceq 958   =/= wne 1588  (/)c0 2283  U.cuni 2507  Ord word 2953  Lim wlim 2955

This theorem is referenced by:  0ellim 3037  dflim3 3124  tz7.44lem1 3933  dfrdg2 3939

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 779  df-cleq 1472  df-ne 1590  df-lim 2959

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