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 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2209   ≠ wne 2420  ∅c0 3520

This proof depends on 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 proof 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 used by:  ne0d  3529  ne0ii  3531  vn0  3532  inelcm  3585  rzal  3625  rexn0  3626  snnzg  3830  prnz  3836  tpnz  3839  brne0  4180  onn0  4545  nn0eln0  4767  ordge1n0im  6709  nnmord  6790  map0g  6969  phpm  7167  fiintim  7238  addclpi  7694  mulclpi  7695  uzn0  9938  iccsupr  10368  pfxn0  11460  ringn0  14365  birthdaylem1g  16087