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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 4992). (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 484 | . . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 ⊆ ∩ 𝐴) | |
2 | df-ne 2942 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
3 | intssuni 4975 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
4 | 2, 3 | sylbir 234 | . . . . . 6 ⊢ (¬ 𝐴 = ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) |
5 | 4 | adantl 483 | . . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∩ 𝐴 ⊆ ∪ 𝐴) |
6 | 1, 5 | eqssd 4000 | . . . 4 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 = ∩ 𝐴) |
7 | 6 | ex 414 | . . 3 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (¬ 𝐴 = ∅ → ∪ 𝐴 = ∩ 𝐴)) |
8 | 7 | orrd 862 | . 2 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴)) |
9 | ssv 4007 | . . . . 5 ⊢ ∪ 𝐴 ⊆ V | |
10 | int0 4967 | . . . . 5 ⊢ ∩ ∅ = V | |
11 | 9, 10 | sseqtrri 4020 | . . . 4 ⊢ ∪ 𝐴 ⊆ ∩ ∅ |
12 | inteq 4954 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
13 | 11, 12 | sseqtrrid 4036 | . . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ⊆ ∩ 𝐴) |
14 | eqimss 4041 | . . 3 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∪ 𝐴 ⊆ ∩ 𝐴) | |
15 | 13, 14 | jaoi 856 | . 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 397 ∨ wo 846 = wceq 1542 ≠ wne 2941 Vcvv 3475 ⊆ wss 3949 ∅c0 4323 ∪ cuni 4909 ∩ cint 4951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-v 3477 df-dif 3952 df-in 3956 df-ss 3966 df-nul 4324 df-uni 4910 df-int 4952 |
This theorem is referenced by: (None) |
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