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Mirrors > Home > MPE Home > Th. List > issn | Structured version Visualization version GIF version |
Description: A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.) |
Ref | Expression |
---|---|
issn | ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equcom 2022 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)) |
3 | 2 | ralbidv 3173 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = 𝑥)) |
4 | ne0i 4293 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
5 | eqsn 4788 | . . . . 5 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝑥} ↔ ∀𝑦 ∈ 𝐴 𝑦 = 𝑥)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐴 = {𝑥} ↔ ∀𝑦 ∈ 𝐴 𝑦 = 𝑥)) |
7 | 3, 6 | bitr4d 282 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ 𝐴 = {𝑥})) |
8 | sneq 4595 | . . . . 5 ⊢ (𝑧 = 𝑥 → {𝑧} = {𝑥}) | |
9 | 8 | eqeq2d 2749 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐴 = {𝑧} ↔ 𝐴 = {𝑥})) |
10 | 9 | spcegv 3555 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 = {𝑥} → ∃𝑧 𝐴 = {𝑧})) |
11 | 7, 10 | sylbid 239 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧})) |
12 | 11 | rexlimiv 3144 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2942 ∀wral 3063 ∃wrex 3072 ∅c0 4281 {csn 4585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2943 df-ral 3064 df-rex 3073 df-v 3446 df-dif 3912 df-in 3916 df-ss 3926 df-nul 4282 df-sn 4586 |
This theorem is referenced by: n0snor2el 4790 f1cdmsn 7225 |
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