Theorem nlim0 4485

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 3495	. . 3 |- -. (/) e. (/)
2		simp2 1022	. . 3 |- ( ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) ) -> (/) e. (/) )
3	1, 2	mto 666	. 2 |- -. ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) )
4		dflim2 4461	. 2 |- ( Lim (/) <-> ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) ) )
5	3, 4	mtbir 675	1 |- -. Lim (/)

Syntax hints:   -. wn 3   /\ w3a 1002   = wceq 1395   e. wcel 2200   (/)c0 3491   U.cuni 3888   Ord word 4453   Lim wlim 4455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-nul 3492  df-ilim 4460

This theorem is referenced by: (None)