Theorem n0ii 3922

Description: If a set has elements, then it is not empty. Inference associated with n0i 3920. (Contributed by BJ, 15-Jul-2021.)

Hypothesis

Ref	Expression
n0ii.1	|- A e. B

Assertion

Ref	Expression
n0ii	|- -. B = (/)

Proof of Theorem n0ii

Step	Hyp	Ref	Expression
1		n0ii.1	. 2 |- A e. B
2		n0i 3920	. 2 |- ( A e. B -> -. B = (/) )
3	1, 2	ax-mp 5	1 |- -. B = (/)

Syntax hints:   -. wn 3   = 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:  iin0  4839  snsn0non  5846  tfrlem16  7489  pwcdadom  9038  nnunb  11288  hon0  28652  dmadjrnb  28765  bnj98  30937  dvnprodlem3  40163