Theorem ne0d 3516

Description: Deduction form of ne0i 3515. 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 3515	. 2 ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)
3	1, 2	syl 14	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2203   ≠ wne 2412  ∅c0 3508

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-v 2815  df-dif 3213  df-nul 3509

This theorem is referenced by:  fihashelne0d  11160  mndbn0  13644  grpbn0  13743  bln0  15283