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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 4449). (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 472 | . . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 ⊆ ∩ 𝐴) | |
2 | df-ne 2782 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
3 | intssuni 4434 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
4 | 2, 3 | sylbir 224 | . . . . . 6 ⊢ (¬ 𝐴 = ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) |
5 | 4 | adantl 481 | . . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∩ 𝐴 ⊆ ∪ 𝐴) |
6 | 1, 5 | eqssd 3585 | . . . 4 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 = ∩ 𝐴) |
7 | 6 | ex 449 | . . 3 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (¬ 𝐴 = ∅ → ∪ 𝐴 = ∩ 𝐴)) |
8 | 7 | orrd 392 | . 2 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴)) |
9 | ssv 3588 | . . . . 5 ⊢ ∪ 𝐴 ⊆ V | |
10 | int0 4425 | . . . . 5 ⊢ ∩ ∅ = V | |
11 | 9, 10 | sseqtr4i 3601 | . . . 4 ⊢ ∪ 𝐴 ⊆ ∩ ∅ |
12 | inteq 4413 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
13 | 11, 12 | syl5sseqr 3617 | . . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ⊆ ∩ 𝐴) |
14 | eqimss 3620 | . . 3 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∪ 𝐴 ⊆ ∩ 𝐴) | |
15 | 13, 14 | jaoi 393 | . 2 ⊢ ((𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴) → ∪ 𝐴 ⊆ ∩ 𝐴) |
16 | 8, 15 | impbii 198 | 1 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 ↔ (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-uni 4373 df-int 4411 |
This theorem is referenced by: (None) |
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