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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 3230 | . . 3 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅)) | |
2 | un0 3401 | . . 3 ⊢ (∅ ∪ ∅) = ∅ | |
3 | 1, 2 | eqtrdi 2189 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = ∅) |
4 | ssun1 3244 | . . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
5 | sseq2 3126 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐴 ⊆ ∅)) | |
6 | 4, 5 | mpbii 147 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 ⊆ ∅) |
7 | ss0b 3407 | . . . 4 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅) | |
8 | 6, 7 | sylib 121 | . . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 = ∅) |
9 | ssun2 3245 | . . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
10 | sseq2 3126 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐵 ⊆ ∅)) | |
11 | 9, 10 | mpbii 147 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 ⊆ ∅) |
12 | ss0b 3407 | . . . 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 1332 ∪ cun 3074 ⊆ wss 3076 ∅c0 3368 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 |
This theorem is referenced by: undisj1 3425 undisj2 3426 disjpr2 3595 |
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