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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 3149 | . . 3 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅)) | |
2 | un0 3316 | . . 3 ⊢ (∅ ∪ ∅) = ∅ | |
3 | 1, 2 | syl6eq 2136 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = ∅) |
4 | ssun1 3163 | . . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
5 | sseq2 3048 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐴 ⊆ ∅)) | |
6 | 4, 5 | mpbii 146 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 ⊆ ∅) |
7 | ss0b 3322 | . . . 4 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅) | |
8 | 6, 7 | sylib 120 | . . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 = ∅) |
9 | ssun2 3164 | . . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
10 | sseq2 3048 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐵 ⊆ ∅)) | |
11 | 9, 10 | mpbii 146 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 ⊆ ∅) |
12 | ss0b 3322 | . . . 4 ⊢ (𝐵 ⊆ ∅ ↔ 𝐵 = ∅) | |
13 | 11, 12 | sylib 120 | . . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 = ∅) |
14 | 8, 13 | jca 300 | . 2 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅)) |
15 | 3, 14 | impbii 124 | 1 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 = wceq 1289 ∪ cun 2997 ⊆ wss 2999 ∅c0 3286 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 |
This theorem is referenced by: undisj1 3340 undisj2 3341 disjpr2 3506 |
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