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Mirrors > Home > MPE Home > Th. List > moel | Structured version Visualization version GIF version |
Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2154. (Revised by Wolf Lammen, 23-Nov-2024.) |
Ref | Expression |
---|---|
moel | ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.21v 1942 | . . . 4 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦))) | |
2 | impexp 451 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦))) | |
3 | 2 | albii 1822 | . . . 4 ⊢ (∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑥 = 𝑦))) |
4 | df-ral 3069 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦)) | |
5 | 4 | imbi2i 336 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 = 𝑦))) |
6 | 1, 3, 5 | 3bitr4i 303 | . . 3 ⊢ (∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)) |
7 | 6 | albii 1822 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)) |
8 | eleq1w 2821 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
9 | 8 | mo4 2566 | . 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) |
10 | df-ral 3069 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)) | |
11 | 7, 9, 10 | 3bitr4i 303 | 1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1537 ∈ wcel 2106 ∃*wmo 2538 ∀wral 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-mo 2540 df-clel 2816 df-ral 3069 |
This theorem is referenced by: disjnf 30906 isthinc3 46271 isthincd2lem1 46275 mndtcbas2 46337 |
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