Theorem moel 3386

Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2190. (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		eleq1w 2844	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	mo4 2592	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
3		r2al 3197	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
4	2, 3	bitr4i 280	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   ∈ wcel 2141  ∃*wmo 2563  ∀wral 3075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-mo 2565  df-clel 2836  df-ral 3076

This theorem is referenced by:  disjnf  32729  oppcmndclem  49598  isthinc3  50002  isthincd2lem1  50006  termcbasmo  50064  arweuthinc  50110  arweutermc  50111  funcsn  50122  0fucterm  50124  mndtcbas2  50164