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| Description: Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.) | 
| Ref | Expression | 
|---|---|
| pssdifcom2 | ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊊ (𝐶 ∖ 𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssconb 4142 | . . . 4 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐶) → (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵))) | |
| 2 | 1 | ancoms 458 | . . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵))) | 
| 3 | difcom 4489 | . . . . 5 ⊢ ((𝐶 ∖ 𝐴) ⊆ 𝐵 ↔ (𝐶 ∖ 𝐵) ⊆ 𝐴) | |
| 4 | 3 | notbii 320 | . . . 4 ⊢ (¬ (𝐶 ∖ 𝐴) ⊆ 𝐵 ↔ ¬ (𝐶 ∖ 𝐵) ⊆ 𝐴) | 
| 5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (¬ (𝐶 ∖ 𝐴) ⊆ 𝐵 ↔ ¬ (𝐶 ∖ 𝐵) ⊆ 𝐴)) | 
| 6 | 2, 5 | anbi12d 632 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝐵 ⊆ (𝐶 ∖ 𝐴) ∧ ¬ (𝐶 ∖ 𝐴) ⊆ 𝐵) ↔ (𝐴 ⊆ (𝐶 ∖ 𝐵) ∧ ¬ (𝐶 ∖ 𝐵) ⊆ 𝐴))) | 
| 7 | dfpss3 4089 | . 2 ⊢ (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ (𝐵 ⊆ (𝐶 ∖ 𝐴) ∧ ¬ (𝐶 ∖ 𝐴) ⊆ 𝐵)) | |
| 8 | dfpss3 4089 | . 2 ⊢ (𝐴 ⊊ (𝐶 ∖ 𝐵) ↔ (𝐴 ⊆ (𝐶 ∖ 𝐵) ∧ ¬ (𝐶 ∖ 𝐵) ⊆ 𝐴)) | |
| 9 | 6, 7, 8 | 3bitr4g 314 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊊ (𝐶 ∖ 𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∖ cdif 3948 ⊆ wss 3951 ⊊ wpss 3952 | 
| 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-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-pss 3971 | 
| This theorem is referenced by: fin2i2 10358 | 
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