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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 | 1 | snex 4203 | . . 3 ⊢ {𝐴} ∈ V |
3 | eleq2 2253 | . . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴})) | |
4 | 1 | snid 3638 | . . . 4 ⊢ 𝐴 ∈ {𝐴} |
5 | 3, 4 | intmin3 3886 | . . 3 ⊢ ({𝐴} ∈ V → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴}) |
6 | 2, 5 | ax-mp 5 | . 2 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴} |
7 | 1 | elintab 3870 | . . . 4 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)) |
8 | id 19 | . . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥) | |
9 | 7, 8 | mpgbir 1464 | . . 3 ⊢ 𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} |
10 | snssi 3751 | . . 3 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} |
12 | 6, 11 | eqssi 3186 | 1 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴} |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 {cab 2175 Vcvv 2752 ⊆ wss 3144 {csn 3607 ∩ cint 3859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-int 3860 |
This theorem is referenced by: (None) |
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