Theorem moel 3365

Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2168. (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 2823	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	mo4 2570	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
3		r2al 3176	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
4	2, 3	bitr4i 279	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   ∈ wcel 2119  ∃*wmo 2541  ∀wral 3054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-mo 2543  df-clel 2815  df-ral 3055

This theorem is referenced by:  disjnf  32666  oppcmndclem  49514  isthinc3  49918  isthincd2lem1  49922  termcbasmo  49980  arweuthinc  50026  arweutermc  50027  funcsn  50038  0fucterm  50040  mndtcbas2  50080