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| Mirrors > Home > MPE Home > Th. List > uni0b | Structured version Visualization version GIF version | ||
| Description: The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.) |
| Ref | Expression |
|---|---|
| uni0b | ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | velsn 4642 | . . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
| 2 | 1 | ralbii 3093 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
| 3 | dfss3 3972 | . 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅}) | |
| 4 | neq0 4352 | . . . 4 ⊢ (¬ ∪ 𝐴 = ∅ ↔ ∃𝑦 𝑦 ∈ ∪ 𝐴) | |
| 5 | rexcom4 3288 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) | |
| 6 | neq0 4352 | . . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥) | |
| 7 | 6 | rexbii 3094 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥) |
| 8 | eluni2 4911 | . . . . . 6 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) | |
| 9 | 8 | exbii 1848 | . . . . 5 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) |
| 10 | 5, 7, 9 | 3bitr4ri 304 | . . . 4 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅) |
| 11 | rexnal 3100 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 = ∅) | |
| 12 | 4, 10, 11 | 3bitri 297 | . . 3 ⊢ (¬ ∪ 𝐴 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
| 13 | 12 | con4bii 321 | . 2 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
| 14 | 2, 3, 13 | 3bitr4ri 304 | 1 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅}) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∅c0 4333 {csn 4626 ∪ cuni 4907 |
| 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-11 2157 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 df-sn 4627 df-uni 4908 |
| This theorem is referenced by: uni0c 4934 uni0 4935 fin1a2lem11 10450 zornn0g 10545 0top 22990 filconn 23891 alexsubALTlem2 24056 ordcmp 36448 unisn0 45059 |
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