Theorem ne0i 3515

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2830. (Contributed by NM, 31-Dec-1993.)

Assertion

Ref	Expression
ne0i	⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3514	. 2 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅)
2	1	neneqad 2491	1 ⊢ (𝐵 ∈ 𝐴 → 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:  ne0d  3516  ne0ii  3518  vn0  3519  inelcm  3569  rzal  3607  rexn0  3608  snnzg  3809  prnz  3815  tpnz  3818  brne0  4159  onn0  4521  nn0eln0  4742  ordge1n0im  6669  nnmord  6750  map0g  6922  phpm  7120  fiintim  7191  addclpi  7642  mulclpi  7643  uzn0  9870  iccsupr  10299  pfxn0  11380  ringn0  14204