Theorem ne0i 3499

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2817. (Contributed by NM, 31-Dec-1993.)

Assertion

Ref	Expression
ne0i	⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3498	. 2 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅)
2	1	neneqad 2479	1 ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2200   ≠ wne 2400  ∅c0 3492

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 2802  df-dif 3200  df-nul 3493

This theorem is referenced by:  ne0d  3500  ne0ii  3502  vn0  3503  inelcm  3553  rzal  3590  rexn0  3591  snnzg  3787  prnz  3793  tpnz  3796  brne0  4136  onn0  4495  nn0eln0  4716  ordge1n0im  6599  nnmord  6680  map0g  6852  phpm  7047  fiintim  7116  addclpi  7537  mulclpi  7538  uzn0  9762  iccsupr  10191  pfxn0  11259  ringn0  14063