Theorem moel 29899

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 3426	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦))
2		df-ral 3095	. . 3 ⊢ (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦))
3	2	ralbii 3162	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦))
4		alcom 2152	. . 3 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
5		eleq1w 2842	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
6	5	mo4 2585	. . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
7		df-ral 3095	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦)))
8		impexp 443	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦)))
9	8	albii 1863	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦)))
10	7, 9	bitr4i 270	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
11	10	albii 1863	. . 3 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
12	4, 6, 11	3bitr4i 295	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦))
13	1, 3, 12	3bitr4ri 296	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599   ∈ wcel 2107  ∃*wmo 2549  ∀wral 3090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-v 3400

This theorem is referenced by:  disjnf  29951