Theorem nlim0 4429

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 3454	. . 3 |- -. (/) e. (/)
2		simp2 1000	. . 3 |- ( ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) ) -> (/) e. (/) )
3	1, 2	mto 663	. 2 |- -. ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) )
4		dflim2 4405	. 2 |- ( Lim (/) <-> ( Ord (/) /\ (/) e. (/) /\ (/) = U. (/) ) )
5	3, 4	mtbir 672	1 |- -. Lim (/)

Syntax hints:   -. wn 3   /\ w3a 980   = wceq 1364   e. wcel 2167   (/)c0 3450   U.cuni 3839   Ord word 4397   Lim wlim 4399

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-nul 3451  df-ilim 4404

This theorem is referenced by: (None)