Theorem neik0imk0p 38334

Description: Kuratowski's K0 axiom implies K0'. Neighborhood version. Also a proof the dual KA axiom imples KA' when considering the convergents. (Contributed by RP, 28-Jun-2021.)

Assertion

Ref	Expression
neik0imk0p	|- ( A. x e. B B e. ( N ` x ) -> A. x e. B ( N ` x ) =/= (/) )

Proof of Theorem neik0imk0p

Step	Hyp	Ref	Expression
1		ne0i 3921	. 2 |- ( B e. ( N ` x ) -> ( N ` x ) =/= (/) )
2	1	ralimi 2952	1 |- ( A. x e. B B e. ( N ` x ) -> A. x e. B ( N ` x ) =/= (/) )

Syntax hints:   -> wi 4   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)c0 3915   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)