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Mirrors > Home > NFE Home > Th. List > unissint | 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 3963). (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 443 | . . . . 5 ⊢ ((∪A ⊆ ∩A ∧ ¬ A = ∅) → ∪A ⊆ ∩A) | |
2 | df-ne 2518 | . . . . . . 7 ⊢ (A ≠ ∅ ↔ ¬ A = ∅) | |
3 | intssuni 3948 | . . . . . . 7 ⊢ (A ≠ ∅ → ∩A ⊆ ∪A) | |
4 | 2, 3 | sylbir 204 | . . . . . 6 ⊢ (¬ A = ∅ → ∩A ⊆ ∪A) |
5 | 4 | adantl 452 | . . . . 5 ⊢ ((∪A ⊆ ∩A ∧ ¬ A = ∅) → ∩A ⊆ ∪A) |
6 | 1, 5 | eqssd 3289 | . . . 4 ⊢ ((∪A ⊆ ∩A ∧ ¬ A = ∅) → ∪A = ∩A) |
7 | 6 | ex 423 | . . 3 ⊢ (∪A ⊆ ∩A → (¬ A = ∅ → ∪A = ∩A)) |
8 | 7 | orrd 367 | . 2 ⊢ (∪A ⊆ ∩A → (A = ∅ ∨ ∪A = ∩A)) |
9 | ssv 3291 | . . . . 5 ⊢ ∪A ⊆ V | |
10 | int0 3940 | . . . . 5 ⊢ ∩∅ = V | |
11 | 9, 10 | sseqtr4i 3304 | . . . 4 ⊢ ∪A ⊆ ∩∅ |
12 | inteq 3929 | . . . 4 ⊢ (A = ∅ → ∩A = ∩∅) | |
13 | 11, 12 | syl5sseqr 3320 | . . 3 ⊢ (A = ∅ → ∪A ⊆ ∩A) |
14 | eqimss 3323 | . . 3 ⊢ (∪A = ∩A → ∪A ⊆ ∩A) | |
15 | 13, 14 | jaoi 368 | . 2 ⊢ ((A = ∅ ∨ ∪A = ∩A) → ∪A ⊆ ∩A) |
16 | 8, 15 | impbii 180 | 1 ⊢ (∪A ⊆ ∩A ↔ (A = ∅ ∨ ∪A = ∩A)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 176 ∨ wo 357 ∧ wa 358 = wceq 1642 ≠ wne 2516 Vcvv 2859 ⊆ wss 3257 ∅c0 3550 ∪cuni 3891 ∩cint 3926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 df-uni 3892 df-int 3927 |
This theorem is referenced by: (None) |
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