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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 3935 | . . . 4 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
3 | 1, 2 | ax-mp 7 | . . 3 ⊢ {𝐴} ∈ V |
4 | eleq2 2101 | . . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴})) | |
5 | 1 | snid 3402 | . . . 4 ⊢ 𝐴 ∈ {𝐴} |
6 | 4, 5 | intmin3 3642 | . . 3 ⊢ ({𝐴} ∈ V → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴}) |
7 | 3, 6 | ax-mp 7 | . 2 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴} |
8 | 1 | elintab 3626 | . . . 4 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)) |
9 | id 19 | . . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥) | |
10 | 8, 9 | mpgbir 1342 | . . 3 ⊢ 𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} |
11 | snssi 3508 | . . 3 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}) | |
12 | 10, 11 | ax-mp 7 | . 2 ⊢ {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} |
13 | 7, 12 | eqssi 2961 | 1 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴} |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 {cab 2026 Vcvv 2557 ⊆ wss 2917 {csn 3375 ∩ cint 3615 |
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-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 |
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-v 2559 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-int 3616 |
This theorem is referenced by: (None) |
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