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Mirrors > Home > MPE Home > Th. List > ssunsn | Structured version Visualization version GIF version |
Description: Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.) |
Ref | Expression |
---|---|
ssunsn | ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssunsn2 4760 | . 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∨ ((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})))) | |
2 | ancom 463 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
3 | eqss 3982 | . . . 4 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
4 | 2, 3 | bitr4i 280 | . . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 = 𝐵) |
5 | ancom 463 | . . . 4 ⊢ (((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 ⊆ (𝐵 ∪ {𝐶}) ∧ (𝐵 ∪ {𝐶}) ⊆ 𝐴)) | |
6 | eqss 3982 | . . . 4 ⊢ (𝐴 = (𝐵 ∪ {𝐶}) ↔ (𝐴 ⊆ (𝐵 ∪ {𝐶}) ∧ (𝐵 ∪ {𝐶}) ⊆ 𝐴)) | |
7 | 5, 6 | bitr4i 280 | . . 3 ⊢ (((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ 𝐴 = (𝐵 ∪ {𝐶})) |
8 | 4, 7 | orbi12i 911 | . 2 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∨ ((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶}))) ↔ (𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶}))) |
9 | 1, 8 | bitri 277 | 1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶}))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∪ cun 3934 ⊆ wss 3936 {csn 4567 |
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-3an 1085 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-ral 3143 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-sn 4568 |
This theorem is referenced by: ssunpr 4765 |
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