Theorem neq0r 3439

Description: An inhabited class is nonempty. See n0rf 3437 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 3438	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	neneqd 2368	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∅c0 3424

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2741  df-dif 3133  df-nul 3425

This theorem is referenced by:  exmidsssn  4204  fzn  10045