Theorem n0rex 4285

Description: There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020.)

Assertion

Ref	Expression
n0rex	⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem n0rex

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)
2	1	ancli 548	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
3	2	eximi 1838	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
4		n0 4277	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		df-rex 3069	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
6	3, 4, 5	3imtr4i 291	1 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064  ∅c0 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-ne 2943  df-rex 3069  df-dif 3886  df-nul 4254

This theorem is referenced by:  ssn0rex  4286