Theorem n0ii 3455

Description: If a class has elements, then it is not empty. Inference associated with n0i 3452. (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 3452	. 2 |- ( A e. B -> -. B = (/) )
3	1, 2	ax-mp 5	1 |- -. B = (/)

Syntax hints:   -. wn 3   = wceq 1364   e. wcel 2164   (/)c0 3446

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-nul 3447

This theorem is referenced by: (None)