Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3957 | . . 3 ⊢ {𝐴} ∈ V |
3 | eleq2 2142 | . . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴})) | |
4 | 1 | snid 3425 | . . . 4 ⊢ 𝐴 ∈ {𝐴} |
5 | 3, 4 | intmin3 3663 | . . 3 ⊢ ({𝐴} ∈ V → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴}) |
6 | 2, 5 | ax-mp 7 | . 2 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴} |
7 | 1 | elintab 3647 | . . . 4 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)) |
8 | id 19 | . . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥) | |
9 | 7, 8 | mpgbir 1382 | . . 3 ⊢ 𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} |
10 | snssi 3529 | . . 3 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}) | |
11 | 9, 10 | ax-mp 7 | . 2 ⊢ {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} |
12 | 6, 11 | eqssi 3015 | 1 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴} |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 {cab 2067 Vcvv 2601 ⊆ wss 2973 {csn 3398 ∩ cint 3636 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-int 3637 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |