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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 2028 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)) |
3 | 2 | ralbidv 3108 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = 𝑥)) |
4 | ne0i 4235 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
5 | eqsn 4728 | . . . . 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 285 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ 𝐴 = {𝑥})) |
8 | sneq 4537 | . . . . 5 ⊢ (𝑧 = 𝑥 → {𝑧} = {𝑥}) | |
9 | 8 | eqeq2d 2747 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐴 = {𝑧} ↔ 𝐴 = {𝑥})) |
10 | 9 | spcegv 3502 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 = {𝑥} → ∃𝑧 𝐴 = {𝑧})) |
11 | 7, 10 | sylbid 243 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧})) |
12 | 11 | rexlimiv 3189 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 ∃wrex 3052 ∅c0 4223 {csn 4527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-v 3400 df-dif 3856 df-in 3860 df-ss 3870 df-nul 4224 df-sn 4528 |
This theorem is referenced by: n0snor2el 4730 |
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