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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-disj2r | Structured version Visualization version GIF version |
Description: Relative version of ssdifin0 4433, allowing a biconditional, and of disj2 4409. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssdifin0 4433 nor disj2 4409. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-disj2r | ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ss 3954 | . . 3 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = (𝐴 ∩ 𝑉)) | |
2 | indif2 4249 | . . . . 5 ⊢ ((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = (((𝐴 ∩ 𝑉) ∩ 𝑉) ∖ 𝐵) | |
3 | inss1 4207 | . . . . . . 7 ⊢ ((𝐴 ∩ 𝑉) ∩ 𝑉) ⊆ (𝐴 ∩ 𝑉) | |
4 | ssid 3991 | . . . . . . . 8 ⊢ (𝐴 ∩ 𝑉) ⊆ (𝐴 ∩ 𝑉) | |
5 | inss2 4208 | . . . . . . . 8 ⊢ (𝐴 ∩ 𝑉) ⊆ 𝑉 | |
6 | 4, 5 | ssini 4210 | . . . . . . 7 ⊢ (𝐴 ∩ 𝑉) ⊆ ((𝐴 ∩ 𝑉) ∩ 𝑉) |
7 | 3, 6 | eqssi 3985 | . . . . . 6 ⊢ ((𝐴 ∩ 𝑉) ∩ 𝑉) = (𝐴 ∩ 𝑉) |
8 | 7 | difeq1i 4097 | . . . . 5 ⊢ (((𝐴 ∩ 𝑉) ∩ 𝑉) ∖ 𝐵) = ((𝐴 ∩ 𝑉) ∖ 𝐵) |
9 | 2, 8 | eqtri 2846 | . . . 4 ⊢ ((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = ((𝐴 ∩ 𝑉) ∖ 𝐵) |
10 | 9 | eqeq1i 2828 | . . 3 ⊢ (((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = (𝐴 ∩ 𝑉) ↔ ((𝐴 ∩ 𝑉) ∖ 𝐵) = (𝐴 ∩ 𝑉)) |
11 | eqcom 2830 | . . 3 ⊢ (((𝐴 ∩ 𝑉) ∖ 𝐵) = (𝐴 ∩ 𝑉) ↔ (𝐴 ∩ 𝑉) = ((𝐴 ∩ 𝑉) ∖ 𝐵)) | |
12 | 1, 10, 11 | 3bitri 299 | . 2 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐴 ∩ 𝑉) = ((𝐴 ∩ 𝑉) ∖ 𝐵)) |
13 | disj3 4405 | . 2 ⊢ (((𝐴 ∩ 𝑉) ∩ 𝐵) = ∅ ↔ (𝐴 ∩ 𝑉) = ((𝐴 ∩ 𝑉) ∖ 𝐵)) | |
14 | in32 4200 | . . 3 ⊢ ((𝐴 ∩ 𝑉) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∩ 𝑉) | |
15 | 14 | eqeq1i 2828 | . 2 ⊢ (((𝐴 ∩ 𝑉) ∩ 𝐵) = ∅ ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅) |
16 | 12, 13, 15 | 3bitr2i 301 | 1 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∖ cdif 3935 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 |
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 2795 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 |
This theorem is referenced by: bj-sscon 34343 |
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