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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 2944 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
2 | biorf 935 | . . 3 ⊢ (¬ 𝐴 = ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))) | |
3 | 1, 2 | sylbi 216 | . 2 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))) |
4 | dfss3 3932 | . . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵}) | |
5 | sssn 4786 | . . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})) | |
6 | velsn 4602 | . . . 4 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
7 | 6 | ralbii 3096 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵) |
8 | 4, 5, 7 | 3bitr3i 300 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵) |
9 | 3, 8 | bitrdi 286 | 1 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ⊆ wss 3910 ∅c0 4282 {csn 4586 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-ral 3065 df-v 3447 df-dif 3913 df-in 3917 df-ss 3927 df-nul 4283 df-sn 4587 |
This theorem is referenced by: issn 4790 zornn0g 10440 hashgt12el 14321 hashgt12el2 14322 hashge2el2dif 14378 simpgnideld 19876 lssne0 20409 qtopeu 23065 dimval 32291 dimvalfi 32292 rngoueqz 36390 lmod0rng 46138 |
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