Theorem neq0r 3280

Description: An inhabited class is nonempty. See n0rf 3278 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 3279	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	neneqd 2270	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1285  ∃wex 1422   ∈ wcel 1434  ∅c0 3269

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-v 2614  df-dif 2986  df-nul 3270

This theorem is referenced by:  fzn  9350