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 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵) | |
3 | eqsnd.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
4 | 3 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐵 ∈ 𝐴) |
5 | 2, 4 | eqeltrd 2912 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴) |
6 | 1, 5 | impbida 799 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 = 𝐵)) |
7 | velsn 4576 | . . 3 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
8 | 6, 7 | syl6bbr 291 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝐵})) |
9 | 8 | eqrdv 2818 | 1 ⊢ (𝜑 → 𝐴 = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {csn 4560 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-v 3493 df-sn 4561 |
This theorem is referenced by: lbsdiflsp0 31044 |
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