Theorem ne0d 3416

Description: Deduction form of ne0i 3415. If a class has elements, then it is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypothesis

Ref	Expression
ne0d.1	⊢ (𝜑 → 𝐵 ∈ 𝐴)

Assertion

Ref	Expression
ne0d	⊢ (𝜑 → 𝐴 ≠ ∅)

Proof of Theorem ne0d

Step	Hyp	Ref	Expression
1		ne0d.1	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
2		ne0i 3415	. 2 ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)
3	1, 2	syl 14	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2136   ≠ wne 2336  ∅c0 3409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-v 2728  df-dif 3118  df-nul 3410

This theorem is referenced by:  bln0  13058