Theorem neq0r 3324

Description: An inhabited class is nonempty. See n0rf 3322 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 3323	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	neneqd 2288	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1299  ∃wex 1436   ∈ wcel 1448  ∅c0 3310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-v 2643  df-dif 3023  df-nul 3311

This theorem is referenced by:  exmidsssn  4063  fzn  9663