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Mirrors > Home > MPE Home > Th. List > unissint | Structured version Visualization version GIF version |
Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4946). (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 483 | . . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 ⊆ ∩ 𝐴) | |
2 | df-ne 2942 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
3 | intssuni 4929 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
4 | 2, 3 | sylbir 234 | . . . . . 6 ⊢ (¬ 𝐴 = ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) |
5 | 4 | adantl 482 | . . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∩ 𝐴 ⊆ ∪ 𝐴) |
6 | 1, 5 | eqssd 3959 | . . . 4 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 = ∩ 𝐴) |
7 | 6 | ex 413 | . . 3 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (¬ 𝐴 = ∅ → ∪ 𝐴 = ∩ 𝐴)) |
8 | 7 | orrd 861 | . 2 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴)) |
9 | ssv 3966 | . . . . 5 ⊢ ∪ 𝐴 ⊆ V | |
10 | int0 4921 | . . . . 5 ⊢ ∩ ∅ = V | |
11 | 9, 10 | sseqtrri 3979 | . . . 4 ⊢ ∪ 𝐴 ⊆ ∩ ∅ |
12 | inteq 4908 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
13 | 11, 12 | sseqtrrid 3995 | . . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ⊆ ∩ 𝐴) |
14 | eqimss 3998 | . . 3 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∪ 𝐴 ⊆ ∩ 𝐴) | |
15 | 13, 14 | jaoi 855 | . 2 ⊢ ((𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴) → ∪ 𝐴 ⊆ ∩ 𝐴) |
16 | 8, 15 | impbii 208 | 1 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 ↔ (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ≠ wne 2941 Vcvv 3443 ⊆ wss 3908 ∅c0 4280 ∪ cuni 4863 ∩ cint 4905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-v 3445 df-dif 3911 df-in 3915 df-ss 3925 df-nul 4281 df-uni 4864 df-int 4906 |
This theorem is referenced by: (None) |
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