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Mirrors > Home > MPE Home > Th. List > eqsnd | Structured version Visualization version GIF version |
Description: Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023.) (Proof shortened by SN, 3-Jul-2025.) |
Ref | Expression |
---|---|
eqsnd.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = 𝐵) |
eqsnd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
Ref | Expression |
---|---|
eqsnd | ⊢ (𝜑 → 𝐴 = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsnd.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = 𝐵) | |
2 | 1 | ralrimiva 3144 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 = 𝐵) |
3 | eqsnd.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
4 | 3 | ne0d 4348 | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) |
5 | eqsn 4834 | . . 3 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)) |
7 | 2, 6 | mpbird 257 | 1 ⊢ (𝜑 → 𝐴 = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∅c0 4339 {csn 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-v 3480 df-dif 3966 df-ss 3980 df-nul 4340 df-sn 4632 |
This theorem is referenced by: 0ringidl 33429 lbsdiflsp0 33654 fiabv 42523 |
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