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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 2702 | . . 3 ⊢ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = {𝑥 ∣ 𝐴 ⊆ 𝑥} | |
2 | 1 | inteqi 3770 | . 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} |
3 | intmin2.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | intmin 3786 | . . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴 |
6 | 2, 5 | eqtr3i 2160 | 1 ⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 {cab 2123 {crab 2418 Vcvv 2681 ⊆ wss 3066 ∩ cint 3766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rab 2423 df-v 2683 df-in 3072 df-ss 3079 df-int 3767 |
This theorem is referenced by: (None) |
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