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Mirrors > Home > ILE Home > Th. List > intid | GIF version |
Description: The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.) |
Ref | Expression |
---|---|
intid.1 | ⊢ A ∈ V |
Ref | Expression |
---|---|
intid | ⊢ ∩ {x ∣ A ∈ x} = {A} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intid.1 | . . . 4 ⊢ A ∈ V | |
2 | snexgOLD 3926 | . . . 4 ⊢ (A ∈ V → {A} ∈ V) | |
3 | 1, 2 | ax-mp 7 | . . 3 ⊢ {A} ∈ V |
4 | eleq2 2098 | . . . 4 ⊢ (x = {A} → (A ∈ x ↔ A ∈ {A})) | |
5 | 1 | snid 3394 | . . . 4 ⊢ A ∈ {A} |
6 | 4, 5 | intmin3 3633 | . . 3 ⊢ ({A} ∈ V → ∩ {x ∣ A ∈ x} ⊆ {A}) |
7 | 3, 6 | ax-mp 7 | . 2 ⊢ ∩ {x ∣ A ∈ x} ⊆ {A} |
8 | 1 | elintab 3617 | . . . 4 ⊢ (A ∈ ∩ {x ∣ A ∈ x} ↔ ∀x(A ∈ x → A ∈ x)) |
9 | id 19 | . . . 4 ⊢ (A ∈ x → A ∈ x) | |
10 | 8, 9 | mpgbir 1339 | . . 3 ⊢ A ∈ ∩ {x ∣ A ∈ x} |
11 | snssi 3499 | . . 3 ⊢ (A ∈ ∩ {x ∣ A ∈ x} → {A} ⊆ ∩ {x ∣ A ∈ x}) | |
12 | 10, 11 | ax-mp 7 | . 2 ⊢ {A} ⊆ ∩ {x ∣ A ∈ x} |
13 | 7, 12 | eqssi 2955 | 1 ⊢ ∩ {x ∣ A ∈ x} = {A} |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 {cab 2023 Vcvv 2551 ⊆ wss 2911 {csn 3367 ∩ cint 3606 |
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-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 |
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-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-int 3607 |
This theorem is referenced by: (None) |
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