Theorem nlim0 4491

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 3498	. . 3 |- -. (/) e. (/)
2		simp2 1024	. . 3 |- ( ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) ) -> (/) e. (/) )
3	1, 2	mto 668	. 2 |- -. ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) )
4		dflim2 4467	. 2 |- ( Lim (/) <-> ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) ) )
5	3, 4	mtbir 677	1 |- -. Lim (/)

Syntax hints:   -. wn 3   /\ w3a 1004   = wceq 1397   e. wcel 2202   (/)c0 3494   U.cuni 3893   Ord word 4459   Lim wlim 4461

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-nul 3495  df-ilim 4466

This theorem is referenced by: (None)