Theorem ne0i 3503

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2820. (Contributed by NM, 31-Dec-1993.)

Assertion

Ref	Expression
ne0i	⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3502	. 2 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅)
2	1	neneqad 2482	1 ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2202   ≠ wne 2403  ∅c0 3496

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-v 2805  df-dif 3203  df-nul 3497

This theorem is referenced by:  ne0d  3504  ne0ii  3506  vn0  3507  inelcm  3557  rzal  3594  rexn0  3595  snnzg  3793  prnz  3799  tpnz  3802  brne0  4143  onn0  4503  nn0eln0  4724  ordge1n0im  6647  nnmord  6728  map0g  6900  phpm  7095  fiintim  7166  addclpi  7590  mulclpi  7591  uzn0  9816  iccsupr  10245  pfxn0  11318  ringn0  14137