Theorem ne0i 3498

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2816. (Contributed by NM, 31-Dec-1993.)

Assertion

Ref	Expression
ne0i	⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3497	. 2 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅)
2	1	neneqad 2479	1 ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2200   ≠ wne 2400  ∅c0 3491

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-dif 3199  df-nul 3492

This theorem is referenced by:  ne0d  3499  ne0ii  3501  vn0  3502  inelcm  3552  rzal  3589  rexn0  3590  snnzg  3783  prnz  3789  tpnz  3792  brne0  4132  onn0  4490  nn0eln0  4711  ordge1n0im  6580  nnmord  6661  map0g  6833  phpm  7023  fiintim  7089  addclpi  7510  mulclpi  7511  uzn0  9734  iccsupr  10158  pfxn0  11215  ringn0  14018