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Mirrors > Home > MPE Home > Th. List > inssdif0 | Structured version Visualization version GIF version |
Description: Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) |
Ref | Expression |
---|---|
inssdif0 | ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 4169 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | 1 | imbi1i 352 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)) |
3 | iman 404 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) ↔ ¬ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶)) | |
4 | 2, 3 | bitri 277 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶) ↔ ¬ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶)) |
5 | eldif 3946 | . . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) | |
6 | 5 | anbi2i 624 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))) |
7 | elin 4169 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∖ 𝐶))) | |
8 | anass 471 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))) | |
9 | 6, 7, 8 | 3bitr4ri 306 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶))) |
10 | 4, 9 | xchbinx 336 | . . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶) ↔ ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶))) |
11 | 10 | albii 1820 | . 2 ⊢ (∀𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶) ↔ ∀𝑥 ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶))) |
12 | dfss2 3955 | . 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶)) | |
13 | eq0 4308 | . 2 ⊢ ((𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶))) | |
14 | 11, 12, 13 | 3bitr4i 305 | 1 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3939 df-in 3943 df-ss 3952 df-nul 4292 |
This theorem is referenced by: disjdif 4421 inf3lem3 9093 ssfin4 9732 isnrm2 21966 1stccnp 22070 llycmpkgen2 22158 ufileu 22527 fclscf 22633 flimfnfcls 22636 inindif 30278 opnbnd 33673 diophrw 39376 setindtr 39641 |
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