|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > un00 | Structured version Visualization version 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 4163 | . . 3 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅)) | |
| 2 | un0 4394 | . . 3 ⊢ (∅ ∪ ∅) = ∅ | |
| 3 | 1, 2 | eqtrdi 2793 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = ∅) | 
| 4 | ssun1 4178 | . . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
| 5 | sseq2 4010 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐴 ⊆ ∅)) | |
| 6 | 4, 5 | mpbii 233 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 ⊆ ∅) | 
| 7 | ss0b 4401 | . . . 4 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅) | |
| 8 | 6, 7 | sylib 218 | . . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 = ∅) | 
| 9 | ssun2 4179 | . . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
| 10 | sseq2 4010 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐵 ⊆ ∅)) | |
| 11 | 9, 10 | mpbii 233 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 ⊆ ∅) | 
| 12 | ss0b 4401 | . . . 4 ⊢ (𝐵 ⊆ ∅ ↔ 𝐵 = ∅) | |
| 13 | 11, 12 | sylib 218 | . . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 = ∅) | 
| 14 | 8, 13 | jca 511 | . 2 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅)) | 
| 15 | 3, 14 | impbii 209 | 1 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 | 
| This theorem is referenced by: undisj1 4462 undisj2 4463 disjpr2 4713 rankxplim3 9921 ssxr 11330 rpnnen2lem12 16261 wwlksnext 29913 asindmre 37710 tfsconcat00 43360 iunrelexp0 43715 uneqsn 44038 | 
| Copyright terms: Public domain | W3C validator |