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Mirrors > Home > ILE Home > Th. List > un00 | GIF version |
Description: Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.) |
Ref | Expression |
---|---|
un00 | ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq12 3276 | . . 3 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅)) | |
2 | un0 3447 | . . 3 ⊢ (∅ ∪ ∅) = ∅ | |
3 | 1, 2 | eqtrdi 2219 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = ∅) |
4 | ssun1 3290 | . . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
5 | sseq2 3171 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐴 ⊆ ∅)) | |
6 | 4, 5 | mpbii 147 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 ⊆ ∅) |
7 | ss0b 3453 | . . . 4 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅) | |
8 | 6, 7 | sylib 121 | . . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 = ∅) |
9 | ssun2 3291 | . . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
10 | sseq2 3171 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐵 ⊆ ∅)) | |
11 | 9, 10 | mpbii 147 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 ⊆ ∅) |
12 | ss0b 3453 | . . . 4 ⊢ (𝐵 ⊆ ∅ ↔ 𝐵 = ∅) | |
13 | 11, 12 | sylib 121 | . . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 = ∅) |
14 | 8, 13 | jca 304 | . 2 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅)) |
15 | 3, 14 | impbii 125 | 1 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1348 ∪ cun 3119 ⊆ wss 3121 ∅c0 3414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 |
This theorem is referenced by: undisj1 3471 undisj2 3472 disjpr2 3645 |
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