Theorem n0rex 40310

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 572	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
3	2	eximi 1752	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
4		n0 3890	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		df-rex 2902	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
6	3, 4, 5	3imtr4i 280	1 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rex 2902  df-v 3175  df-dif 3543  df-nul 3875

This theorem is referenced by:  ssn0rex  40311