Theorem ne0i 3466

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2787. (Contributed by NM, 31-Dec-1993.)

Assertion

Ref	Expression
ne0i	⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3465	. 2 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅)
2	1	neneqad 2454	1 ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2175   ≠ wne 2375  ∅c0 3459

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-v 2773  df-dif 3167  df-nul 3460

This theorem is referenced by:  ne0d  3467  ne0ii  3469  vn0  3470  inelcm  3520  rzal  3557  rexn0  3558  snnzg  3749  prnz  3754  tpnz  3757  brne0  4092  onn0  4446  nn0eln0  4667  ordge1n0im  6521  nnmord  6602  map0g  6774  phpm  6961  fiintim  7027  addclpi  7439  mulclpi  7440  uzn0  9663  iccsupr  10087  ringn0  13764