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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 3930 | . . . . 5 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) | |
2 | 1 | imbi2i 336 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) |
3 | 2 | albii 1822 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) |
4 | jcab 519 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))) | |
5 | 4 | albii 1822 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) ↔ ∀𝑥((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))) |
6 | 19.26 1874 | . . 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 298 | . 2 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶))) |
8 | dfss2 3934 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
9 | dfss2 3934 | . . 3 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) | |
10 | 8, 9 | anbi12i 628 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))) |
11 | dfss2 3934 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∩ 𝐶) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶))) | |
12 | 7, 10, 11 | 3bitr4i 303 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 ∈ wcel 2107 ∩ cin 3913 ⊆ wss 3914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3449 df-in 3921 df-ss 3931 |
This theorem is referenced by: ssini 4195 ssind 4196 uneqin 4242 disjpss 4424 trin 5238 pwin 5531 fin 6726 frrlem4 8224 frrlem13 8233 wfrlem4OLD 8262 epfrs 9675 tcmin 9685 resscntz 19120 subgdmdprd 19821 tgval 22328 eltg3i 22334 innei 22499 cnprest2 22664 subislly 22855 lly1stc 22870 xkohaus 23027 xkoinjcn 23061 opnfbas 23216 supfil 23269 rnelfm 23327 tsmsres 23518 restmetu 23949 chabs2 30508 cmbr4i 30592 pjin3i 31185 mdbr2 31287 dmdbr2 31294 dmdbr5 31299 mdslle1i 31308 mdslle2i 31309 mdslj1i 31310 mdslj2i 31311 mdsl2i 31313 mdslmd1lem1 31316 mdslmd1lem2 31317 mdslmd1i 31320 mdslmd3i 31323 hatomistici 31353 chrelat2i 31356 cvexchlem 31359 mdsymlem1 31394 mdsymlem3 31396 mdsymlem6 31399 dmdbr5ati 31413 pnfneige0 32596 ballotlem2 33152 iccllysconn 33908 heibor1lem 36318 relssinxpdmrn 36860 dochexmidlem1 39973 superficl 41931 k0004lem1 42511 ismnushort 42673 |
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