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Mirrors > Home > MPE Home > Th. List > pssdifcom2 | Structured version Visualization version GIF version |
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 4114 | . . . 4 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐶) → (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵))) | |
2 | 1 | ancoms 461 | . . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵))) |
3 | difcom 4434 | . . . . 5 ⊢ ((𝐶 ∖ 𝐴) ⊆ 𝐵 ↔ (𝐶 ∖ 𝐵) ⊆ 𝐴) | |
4 | 3 | notbii 322 | . . . 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 4063 | . 2 ⊢ (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ (𝐵 ⊆ (𝐶 ∖ 𝐴) ∧ ¬ (𝐶 ∖ 𝐴) ⊆ 𝐵)) | |
8 | dfpss3 4063 | . 2 ⊢ (𝐴 ⊊ (𝐶 ∖ 𝐵) ↔ (𝐴 ⊆ (𝐶 ∖ 𝐵) ∧ ¬ (𝐶 ∖ 𝐵) ⊆ 𝐴)) | |
9 | 6, 7, 8 | 3bitr4g 316 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊊ (𝐶 ∖ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∖ cdif 3933 ⊆ wss 3936 ⊊ wpss 3937 |
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-ne 3017 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 |
This theorem is referenced by: fin2i2 9740 |
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