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Mirrors > Home > ILE Home > Th. List > eqsnm | GIF version |
Description: Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.) |
Ref | Expression |
---|---|
eqsnm | ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss3 2935 | . . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵}) | |
2 | velsn 3392 | . . . 4 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
3 | 2 | ralbii 2330 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵) |
4 | 1, 3 | bitri 173 | . 2 ⊢ (𝐴 ⊆ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵) |
5 | sssnm 3525 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵})) | |
6 | 4, 5 | syl5rbbr 184 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∀wral 2306 ⊆ wss 2917 {csn 3375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-in 2924 df-ss 2931 df-sn 3381 |
This theorem is referenced by: (None) |
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