Theorem ne0i 3519

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2832. (Contributed by NM, 31-Dec-1993.)

Assertion

Ref	Expression
ne0i	⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3518	. 2 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅)
2	1	neneqad 2493	1 ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2205   ≠ wne 2414  ∅c0 3512

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-v 2817  df-dif 3216  df-nul 3513

This theorem is referenced by:  ne0d  3520  ne0ii  3522  vn0  3523  inelcm  3573  rzal  3611  rexn0  3612  snnzg  3814  prnz  3820  tpnz  3823  brne0  4164  onn0  4526  nn0eln0  4747  ordge1n0im  6682  nnmord  6763  map0g  6935  phpm  7133  fiintim  7204  addclpi  7658  mulclpi  7659  uzn0  9888  iccsupr  10318  pfxn0  11405  ringn0  14303