Theorem n0rex 3935

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 574	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
3	2	eximi 1762	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
4		n0 3931	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		df-rex 2918	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
6	3, 4, 5	3imtr4i 281	1 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 384  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rex 2918  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  ssn0rex  3936