Theorem n0moeu 4341

Description: A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)

Assertion

Ref	Expression
n0moeu	⊢ (𝐴 ≠ ∅ → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 𝑥 ∈ 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem n0moeu

Step	Hyp	Ref	Expression
1		n0 4335	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2	1	biimpi 216	. . 3 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
3	2	biantrurd 532	. 2 ⊢ (𝐴 ≠ ∅ → (∃*𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃*𝑥 𝑥 ∈ 𝐴)))
4		df-eu 2567	. 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃*𝑥 𝑥 ∈ 𝐴))
5	3, 4	bitr4di 289	1 ⊢ (𝐴 ≠ ∅ → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 𝑥 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1778   ∈ wcel 2107  ∃*wmo 2536  ∃!weu 2566   ≠ wne 2931  ∅c0 4315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-eu 2567  df-clab 2713  df-cleq 2726  df-ne 2932  df-dif 3936  df-nul 4316

This theorem is referenced by:  minveclem4a  25419