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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 2708 |
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 2715 df-cleq 2729 df-clel 2815 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 7728 onsucmin 7748 rankonidlem 9722 rankval4 9761 harsucnn 9892 mrcid 17453 lspid 20396 aspid 21231 cldcls 22345 spanid 30118 chsupid 30183 fldgenidfld 31911 naddid1 34239 naddasslem1 34248 naddasslem2 34249 igenidl2 36462 pclidN 38297 diaocN 39526 onuniintrab 41469 topclat 46924 toplatlub 46926 toplatjoin 46928 |
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