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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 2021 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)) |
3 | 2 | ralbidv 3197 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = 𝑥)) |
4 | ne0i 4299 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
5 | eqsn 4755 | . . . . 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 284 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ 𝐴 = {𝑥})) |
8 | sneq 4570 | . . . . 5 ⊢ (𝑧 = 𝑥 → {𝑧} = {𝑥}) | |
9 | 8 | eqeq2d 2832 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐴 = {𝑧} ↔ 𝐴 = {𝑥})) |
10 | 9 | spcegv 3596 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 = {𝑥} → ∃𝑧 𝐴 = {𝑧})) |
11 | 7, 10 | sylbid 242 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧})) |
12 | 11 | rexlimiv 3280 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∅c0 4290 {csn 4560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-v 3496 df-dif 3938 df-in 3942 df-ss 3951 df-nul 4291 df-sn 4561 |
This theorem is referenced by: n0snor2el 4757 |
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