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| Mirrors > Home > MPE Home > Th. List > elsn2g | Structured version Visualization version GIF version | ||
| Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that 𝐵, rather than 𝐴, be a set. (Contributed by NM, 28-Oct-2003.) |
| Ref | Expression |
|---|---|
| elsn2g | ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elsni 4643 | . 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | |
| 2 | snidg 4660 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵}) | |
| 3 | eleq1 2829 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝐵} ↔ 𝐵 ∈ {𝐵})) | |
| 4 | 2, 3 | syl5ibrcom 247 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ∈ {𝐵})) |
| 5 | 1, 4 | impbid2 226 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 {csn 4626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sn 4627 |
| This theorem is referenced by: elsn2 4665 mptiniseg 6259 elsuc2g 6453 extmptsuppeq 8213 fzosplitsni 13817 1nsgtrivd 19192 limcco 25928 ply1termlem 26242 mptprop 32707 bj-elsn12g 37061 elpmapat 39766 stirlinglem8 46096 dirkercncflem2 46119 clnbgrel 47815 |
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