Theorem nel02 34097

Description: The empty set has no elements. (Contributed by Peter Mazsa, 4-Jan-2018.)

Assertion

Ref	Expression
nel02	|- ( A = (/) -> -. B e. A )

Proof of Theorem nel02

Step	Hyp	Ref	Expression
1		noel 3919	. 2 |- -. B e. (/)
2		eleq2 2690	. 2 |- ( A = (/) -> ( B e. A <-> B e. (/) ) )
3	1, 2	mtbiri 317	1 |- ( A = (/) -> -. B e. A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   (/)c0 3915

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-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)