Theorem ne0i 3528

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2838. (Contributed by NM, 31-Dec-1993.)

Assertion

Ref	Expression
ne0i	⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3527	. 2 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅)
2	1	neneqad 2499	1 ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2209   ≠ wne 2420  ∅c0 3520

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-nul 3521

This theorem is referenced by:  ne0d  3529  ne0ii  3531  vn0  3532  inelcm  3584  rzal  3622  rexn0  3623  snnzg  3825  prnz  3831  tpnz  3834  brne0  4175  onn0  4540  nn0eln0  4762  ordge1n0im  6699  nnmord  6780  map0g  6959  phpm  7157  fiintim  7228  addclpi  7684  mulclpi  7685  uzn0  9917  iccsupr  10347  pfxn0  11438  ringn0  14338