Theorem moel 3325

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 3147	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦))
2		df-ral 3056	. . 3 ⊢ (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦))
3	2	ralbii 3078	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦))
4		alcom 2162	. . 3 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
5		eleq1w 2813	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
6	5	mo4 2565	. . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
7		df-ral 3056	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦)))
8		impexp 454	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦)))
9	8	albii 1827	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦)))
10	7, 9	bitr4i 281	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
11	10	albii 1827	. . 3 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
12	4, 6, 11	3bitr4i 306	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐴 → 𝑥 = 𝑦))
13	1, 3, 12	3bitr4ri 307	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541   ∈ wcel 2112  ∃*wmo 2537  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-11 2160

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-mo 2539  df-clel 2809  df-ral 3056

This theorem is referenced by:  disjnf  30582  isthinc3  45920  isthincd2lem1  45924  mndtcbas2  45984