Theorem neq0r 3476

Description: An inhabited class is nonempty. See n0rf 3474 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)

Assertion

Ref	Expression
neq0r	⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ 𝐴 = ∅)

Distinct variable group:   𝑥,𝐴

Proof of Theorem neq0r

Step	Hyp	Ref	Expression
1		n0r 3475	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	neneqd 2398	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∅c0 3461

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-v 2775  df-dif 3169  df-nul 3462

This theorem is referenced by:  exmidsssn  4250  fzn  10171