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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 3967 | . . . . 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 1819 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) |
| 4 | jcab 517 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))) | |
| 5 | 4 | albii 1819 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) ↔ ∀𝑥((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))) |
| 6 | 19.26 1870 | . . 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 | df-ss 3968 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
| 9 | df-ss 3968 | . . 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 | df-ss 3968 | . 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 206 ∧ wa 395 ∀wal 1538 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-ss 3968 |
| This theorem is referenced by: ssini 4240 ssind 4241 uneqin 4289 disjpss 4461 trin 5271 pwin 5574 fin 6788 frrlem4 8314 frrlem13 8323 wfrlem4OLD 8352 epfrs 9771 tcmin 9781 resscntz 19351 subgdmdprd 20054 tgval 22962 eltg3i 22968 innei 23133 cnprest2 23298 subislly 23489 lly1stc 23504 xkohaus 23661 xkoinjcn 23695 opnfbas 23850 supfil 23903 rnelfm 23961 tsmsres 24152 restmetu 24583 chabs2 31536 cmbr4i 31620 pjin3i 32213 mdbr2 32315 dmdbr2 32322 dmdbr5 32327 mdslle1i 32336 mdslle2i 32337 mdslj1i 32338 mdslj2i 32339 mdsl2i 32341 mdslmd1lem1 32344 mdslmd1lem2 32345 mdslmd1i 32348 mdslmd3i 32351 hatomistici 32381 chrelat2i 32384 cvexchlem 32387 mdsymlem1 32422 mdsymlem3 32424 mdsymlem6 32427 dmdbr5ati 32441 pnfneige0 33950 ballotlem2 34491 iccllysconn 35255 heibor1lem 37816 relssinxpdmrn 38350 dochexmidlem1 41462 superficl 43580 k0004lem1 44160 ismnushort 44320 |
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