Theorem moel 3376

Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2158. (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 2811	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	mo4 2559	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
3		r2al 3173	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
4	2, 3	bitr4i 278	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   ∈ wcel 2109  ∃*wmo 2531  ∀wral 3044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2533  df-clel 2803  df-ral 3045

This theorem is referenced by:  disjnf  32499  oppcmndclem  49003  isthinc3  49407  isthincd2lem1  49411  termcbasmo  49469  arweuthinc  49515  arweutermc  49516  funcsn  49527  0fucterm  49529  mndtcbas2  49569