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Mirrors > Home > MPE Home > Th. List > intmin | Structured version Visualization version GIF version |
Description: Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
intmin | ⊢ (𝐴 ∈ 𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3447 | . . . . 5 ⊢ 𝑦 ∈ V | |
2 | 1 | elintrab 4919 | . . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑥 ∈ 𝐵 (𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥)) |
3 | ssid 3964 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
4 | sseq2 3968 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝐴)) | |
5 | eleq2 2826 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐴)) | |
6 | 4, 5 | imbi12d 344 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥) ↔ (𝐴 ⊆ 𝐴 → 𝑦 ∈ 𝐴))) |
7 | 6 | rspcv 3575 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥) → (𝐴 ⊆ 𝐴 → 𝑦 ∈ 𝐴))) |
8 | 3, 7 | mpii 46 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴)) |
9 | 2, 8 | biimtrid 241 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → 𝑦 ∈ 𝐴)) |
10 | 9 | ssrdv 3948 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ⊆ 𝐴) |
11 | ssintub 4925 | . . 3 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} | |
12 | 11 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}) |
13 | 10, 12 | eqssd 3959 | 1 ⊢ (𝐴 ∈ 𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∀wral 3062 {crab 3405 ⊆ wss 3908 ∩ cint 4905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rab 3406 df-v 3445 df-in 3915 df-ss 3925 df-int 4906 |
This theorem is referenced by: intmin2 4934 ordintdif 6365 uniordint 7732 onsucmin 7752 naddid1 8625 naddasslem1 8634 naddasslem2 8635 rankonidlem 9760 rankval4 9799 harsucnn 9930 mrcid 17485 lspid 20428 aspid 21263 cldcls 22377 spanid 30175 chsupid 30240 fldgenidfld 31968 igenidl2 36491 pclidN 38326 diaocN 39555 onuniintrab 41498 topclat 46955 toplatlub 46957 toplatjoin 46959 |
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