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Mirrors > Home > MPE Home > Th. List > ssin | Structured version Visualization version GIF version |
Description: Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
ssin | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3964 | . . . . 5 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) | |
2 | 1 | imbi2i 335 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) |
3 | 2 | albii 1821 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) |
4 | jcab 518 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))) | |
5 | 4 | albii 1821 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) ↔ ∀𝑥((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))) |
6 | 19.26 1873 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))) | |
7 | 3, 5, 6 | 3bitrri 297 | . 2 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶))) |
8 | dfss2 3968 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
9 | dfss2 3968 | . . 3 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) | |
10 | 8, 9 | anbi12i 627 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))) |
11 | dfss2 3968 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∩ 𝐶) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶))) | |
12 | 7, 10, 11 | 3bitr4i 302 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 ∈ wcel 2106 ∩ cin 3947 ⊆ wss 3948 |
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-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-in 3955 df-ss 3965 |
This theorem is referenced by: ssini 4231 ssind 4232 uneqin 4278 disjpss 4460 trin 5277 pwin 5570 fin 6771 frrlem4 8276 frrlem13 8285 wfrlem4OLD 8314 epfrs 9728 tcmin 9738 resscntz 19199 subgdmdprd 19906 tgval 22465 eltg3i 22471 innei 22636 cnprest2 22801 subislly 22992 lly1stc 23007 xkohaus 23164 xkoinjcn 23198 opnfbas 23353 supfil 23406 rnelfm 23464 tsmsres 23655 restmetu 24086 chabs2 30808 cmbr4i 30892 pjin3i 31485 mdbr2 31587 dmdbr2 31594 dmdbr5 31599 mdslle1i 31608 mdslle2i 31609 mdslj1i 31610 mdslj2i 31611 mdsl2i 31613 mdslmd1lem1 31616 mdslmd1lem2 31617 mdslmd1i 31620 mdslmd3i 31623 hatomistici 31653 chrelat2i 31656 cvexchlem 31659 mdsymlem1 31694 mdsymlem3 31696 mdsymlem6 31699 dmdbr5ati 31713 pnfneige0 33000 ballotlem2 33556 iccllysconn 34310 heibor1lem 36769 relssinxpdmrn 37310 dochexmidlem1 40423 superficl 42406 k0004lem1 42986 ismnushort 43148 |
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