Theorem ne0ii 3501

Description: If a class has elements, then it is nonempty. Inference associated with ne0i 3498. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypothesis

Ref	Expression
n0ii.1	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
ne0ii	⊢ 𝐵 ≠ ∅

Proof of Theorem ne0ii

Step	Hyp	Ref	Expression
1		n0ii.1	. 2 ⊢ 𝐴 ∈ 𝐵
2		ne0i 3498	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅)
3	1, 2	ax-mp 5	1 ⊢ 𝐵 ≠ ∅

Syntax hints:   ∈ wcel 2200   ≠ wne 2400  ∅c0 3491

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-dif 3199  df-nul 3492

This theorem is referenced by:  pw1ne0  7409  sucpw1nel3  7414