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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 3164 | . . 3 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅)) | |
2 | un0 3335 | . . 3 ⊢ (∅ ∪ ∅) = ∅ | |
3 | 1, 2 | syl6eq 2143 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = ∅) |
4 | ssun1 3178 | . . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
5 | sseq2 3063 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐴 ⊆ ∅)) | |
6 | 4, 5 | mpbii 147 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 ⊆ ∅) |
7 | ss0b 3341 | . . . 4 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅) | |
8 | 6, 7 | sylib 121 | . . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 = ∅) |
9 | ssun2 3179 | . . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
10 | sseq2 3063 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐵 ⊆ ∅)) | |
11 | 9, 10 | mpbii 147 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 ⊆ ∅) |
12 | ss0b 3341 | . . . 4 ⊢ (𝐵 ⊆ ∅ ↔ 𝐵 = ∅) | |
13 | 11, 12 | sylib 121 | . . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 = ∅) |
14 | 8, 13 | jca 301 | . 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 1296 ∪ cun 3011 ⊆ wss 3013 ∅c0 3302 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 |
This theorem is referenced by: undisj1 3359 undisj2 3360 disjpr2 3526 |
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