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Mirrors > Home > ILE Home > Th. List > intmin2 | GIF version |
Description: Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.) |
Ref | Expression |
---|---|
intmin2.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
intmin2 | ⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabab 2631 | . . 3 ⊢ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = {𝑥 ∣ 𝐴 ⊆ 𝑥} | |
2 | 1 | inteqi 3666 | . 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} |
3 | intmin2.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | intmin 3682 | . . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴) | |
5 | 3, 4 | ax-mp 7 | . 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴 |
6 | 2, 5 | eqtr3i 2105 | 1 ⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1285 ∈ wcel 1434 {cab 2069 {crab 2357 Vcvv 2612 ⊆ wss 2984 ∩ cint 3662 |
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 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rab 2362 df-v 2614 df-in 2990 df-ss 2997 df-int 3663 |
This theorem is referenced by: (None) |
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