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Mirrors > Home > NFE Home > Th. List > ssunsn | 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 3865 | . 2 ⊢ ((B ⊆ A ∧ A ⊆ (B ∪ {C})) ↔ ((B ⊆ A ∧ A ⊆ B) ∨ ((B ∪ {C}) ⊆ A ∧ A ⊆ (B ∪ {C})))) | |
2 | ancom 437 | . . . 4 ⊢ ((B ⊆ A ∧ A ⊆ B) ↔ (A ⊆ B ∧ B ⊆ A)) | |
3 | eqss 3287 | . . . 4 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A)) | |
4 | 2, 3 | bitr4i 243 | . . 3 ⊢ ((B ⊆ A ∧ A ⊆ B) ↔ A = B) |
5 | ancom 437 | . . . 4 ⊢ (((B ∪ {C}) ⊆ A ∧ A ⊆ (B ∪ {C})) ↔ (A ⊆ (B ∪ {C}) ∧ (B ∪ {C}) ⊆ A)) | |
6 | eqss 3287 | . . . 4 ⊢ (A = (B ∪ {C}) ↔ (A ⊆ (B ∪ {C}) ∧ (B ∪ {C}) ⊆ A)) | |
7 | 5, 6 | bitr4i 243 | . . 3 ⊢ (((B ∪ {C}) ⊆ A ∧ A ⊆ (B ∪ {C})) ↔ A = (B ∪ {C})) |
8 | 4, 7 | orbi12i 507 | . 2 ⊢ (((B ⊆ A ∧ A ⊆ B) ∨ ((B ∪ {C}) ⊆ A ∧ A ⊆ (B ∪ {C}))) ↔ (A = B ∨ A = (B ∪ {C}))) |
9 | 1, 8 | bitri 240 | 1 ⊢ ((B ⊆ A ∧ A ⊆ (B ∪ {C})) ↔ (A = B ∨ A = (B ∪ {C}))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∨ wo 357 ∧ wa 358 = wceq 1642 ∪ cun 3207 ⊆ wss 3257 {csn 3737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 |
This theorem is referenced by: ssunpr 3868 |
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