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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqsnd | Structured version Visualization version GIF version |
Description: Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023.) |
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 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = 𝐵) | |
2 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵) | |
3 | eqsnd.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
4 | 3 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐵 ∈ 𝐴) |
5 | 2, 4 | eqeltrd 2890 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴) |
6 | 1, 5 | impbida 800 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 = 𝐵)) |
7 | velsn 4541 | . . 3 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
8 | 6, 7 | syl6bbr 292 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝐵})) |
9 | 8 | eqrdv 2796 | 1 ⊢ (𝜑 → 𝐴 = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-sn 4526 |
This theorem is referenced by: 0ringidl 31013 lbsdiflsp0 31110 |
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