Theorem moel 3358

Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2154. (Revised by Wolf Lammen, 23-Nov-2024.)

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		19.21v 1942	. . . 4 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦)))
2		impexp 451	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦)))
3	2	albii 1822	. . . 4 ⊢ (∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦)))
4		df-ral 3069	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦))
5	4	imbi2i 336	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦)))
6	1, 3, 5	3bitr4i 303	. . 3 ⊢ (∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝑥 = 𝑦))
7	6	albii 1822	. 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝑥 = 𝑦))
8		eleq1w 2821	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
9	8	mo4 2566	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
10		df-ral 3069	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝑥 = 𝑦))
11	7, 9, 10	3bitr4i 303	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   ∈ wcel 2106  ∃*wmo 2538  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-mo 2540  df-clel 2816  df-ral 3069

This theorem is referenced by:  disjnf  30909  isthinc3  46304  isthincd2lem1  46308  mndtcbas2  46370