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| Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4984). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) | 
| Ref | Expression | 
|---|---|
| unissint | ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 ↔ (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpl 482 | . . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 ⊆ ∩ 𝐴) | |
| 2 | df-ne 2940 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
| 3 | intssuni 4969 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
| 4 | 2, 3 | sylbir 235 | . . . . . 6 ⊢ (¬ 𝐴 = ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | 
| 5 | 4 | adantl 481 | . . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∩ 𝐴 ⊆ ∪ 𝐴) | 
| 6 | 1, 5 | eqssd 4000 | . . . 4 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 = ∩ 𝐴) | 
| 7 | 6 | ex 412 | . . 3 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (¬ 𝐴 = ∅ → ∪ 𝐴 = ∩ 𝐴)) | 
| 8 | 7 | orrd 863 | . 2 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴)) | 
| 9 | ssv 4007 | . . . . 5 ⊢ ∪ 𝐴 ⊆ V | |
| 10 | int0 4961 | . . . . 5 ⊢ ∩ ∅ = V | |
| 11 | 9, 10 | sseqtrri 4032 | . . . 4 ⊢ ∪ 𝐴 ⊆ ∩ ∅ | 
| 12 | inteq 4948 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
| 13 | 11, 12 | sseqtrrid 4026 | . . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ⊆ ∩ 𝐴) | 
| 14 | eqimss 4041 | . . 3 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∪ 𝐴 ⊆ ∩ 𝐴) | |
| 15 | 13, 14 | jaoi 857 | . 2 ⊢ ((𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴) → ∪ 𝐴 ⊆ ∩ 𝐴) | 
| 16 | 8, 15 | impbii 209 | 1 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 ↔ (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1539 ≠ wne 2939 Vcvv 3479 ⊆ wss 3950 ∅c0 4332 ∪ cuni 4906 ∩ cint 4945 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-v 3481 df-dif 3953 df-ss 3967 df-nul 4333 df-uni 4907 df-int 4946 | 
| This theorem is referenced by: (None) | 
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