Theorem emnfil 10476

Description: The empty set is not a filter. Bourbaki TG I.36 def 1 note.

Assertion

Ref	Expression
emnfil	|- -. (/) e. Fil

Proof of Theorem emnfil

Step	Hyp	Ref	Expression
1		noel 2280	. 2 |- -. U.(/) e. (/)
2		eqid 1473	. . 3 |- U.(/) = U.(/)
3	2	filusb 10472	. 2 |- ((/) e. Fil -> U.(/) e. (/))
4	1, 3	mto 106	1 |- -. (/) e. Fil

Syntax hints:  -. wn 2   e. wcel 956  (/)c0 2276  U.cuni 2498  Filcfil 10467

This theorem is referenced by:  cnfilca 10487

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808  df-dif 2045  df-in 2047  df-ss 2049  df-nul 2277  df-uni 2499  df-fil 10468

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