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Mirrors > Home > MPE Home > Th. List > eqsn | Structured version Visualization version GIF version |
Description: Two ways to express that a nonempty set equals a singleton. (Contributed by NM, 15-Dec-2007.) (Proof shortened by JJ, 23-Jul-2021.) |
Ref | Expression |
---|---|
eqsn | ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2988 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
2 | biorf 934 | . . 3 ⊢ (¬ 𝐴 = ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))) | |
3 | 1, 2 | sylbi 220 | . 2 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))) |
4 | dfss3 3903 | . . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵}) | |
5 | sssn 4719 | . . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})) | |
6 | velsn 4541 | . . . 4 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
7 | 6 | ralbii 3133 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵) |
8 | 4, 5, 7 | 3bitr3i 304 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵) |
9 | 3, 8 | syl6bb 290 | 1 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ⊆ wss 3881 ∅c0 4243 {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-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 |
This theorem is referenced by: issn 4723 zornn0g 9916 hashgt12el 13779 hashgt12el2 13780 hashge2el2dif 13834 simpgnideld 19214 lssne0 19715 qtopeu 22321 dimval 31089 dimvalfi 31090 rngoueqz 35378 lmod0rng 44492 |
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