Theorem moel 28707

Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.)

Assertion

Ref	Expression
moel	⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem moel

Step	Hyp	Ref	Expression
1		ralcom4 3197	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦))
2		df-ral 2901	. . 3 ⊢ (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦))
3	2	ralbii 2963	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦))
4		alcom 2024	. . 3 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
5		eleq1 2676	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
6	5	mo4 2505	. . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
7		df-ral 2901	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦)))
8		impexp 461	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦)))
9	8	albii 1737	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦)))
10	7, 9	bitr4i 266	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
11	10	albii 1737	. . 3 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
12	4, 6, 11	3bitr4i 291	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦))
13	1, 3, 12	3bitr4ri 292	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃*wmo 2459  ∀wral 2896

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-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175

This theorem is referenced by:  disjnf  28766