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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 4915). (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 485 | . . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 ⊆ ∩ 𝐴) | |
2 | df-ne 3019 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
3 | intssuni 4900 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
4 | 2, 3 | sylbir 237 | . . . . . 6 ⊢ (¬ 𝐴 = ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) |
5 | 4 | adantl 484 | . . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∩ 𝐴 ⊆ ∪ 𝐴) |
6 | 1, 5 | eqssd 3986 | . . . 4 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 = ∩ 𝐴) |
7 | 6 | ex 415 | . . 3 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (¬ 𝐴 = ∅ → ∪ 𝐴 = ∩ 𝐴)) |
8 | 7 | orrd 859 | . 2 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴)) |
9 | ssv 3993 | . . . . 5 ⊢ ∪ 𝐴 ⊆ V | |
10 | int0 4892 | . . . . 5 ⊢ ∩ ∅ = V | |
11 | 9, 10 | sseqtrri 4006 | . . . 4 ⊢ ∪ 𝐴 ⊆ ∩ ∅ |
12 | inteq 4881 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
13 | 11, 12 | sseqtrrid 4022 | . . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ⊆ ∩ 𝐴) |
14 | eqimss 4025 | . . 3 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∪ 𝐴 ⊆ ∩ 𝐴) | |
15 | 13, 14 | jaoi 853 | . 2 ⊢ ((𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴) → ∪ 𝐴 ⊆ ∩ 𝐴) |
16 | 8, 15 | impbii 211 | 1 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 ↔ (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ≠ wne 3018 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 ∪ cuni 4840 ∩ cint 4878 |
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 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 df-uni 4841 df-int 4879 |
This theorem is referenced by: (None) |
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