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Mirrors > Home > ILE Home > Th. List > elsncg | GIF version |
Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
elsncg | ⊢ (A ∈ 𝑉 → (A ∈ {B} ↔ A = B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2043 | . 2 ⊢ (x = A → (x = B ↔ A = B)) | |
2 | df-sn 3373 | . 2 ⊢ {B} = {x ∣ x = B} | |
3 | 1, 2 | elab2g 2683 | 1 ⊢ (A ∈ 𝑉 → (A ∈ {B} ↔ A = B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ∈ wcel 1390 {csn 3367 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-sn 3373 |
This theorem is referenced by: elsnc 3390 elsni 3391 snidg 3392 eltpg 3407 eldifsn 3486 elsucg 4107 funconstss 5228 fniniseg 5230 fniniseg2 5232 ltxr 8465 elfzp12 8731 1exp 8938 |
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