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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 4573 | . . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
2 | 1 | ralbii 3162 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
3 | dfss3 3953 | . 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅}) | |
4 | neq0 4306 | . . . 4 ⊢ (¬ ∪ 𝐴 = ∅ ↔ ∃𝑦 𝑦 ∈ ∪ 𝐴) | |
5 | rexcom4 3246 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) | |
6 | neq0 4306 | . . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥) | |
7 | 6 | rexbii 3244 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥) |
8 | eluni2 4834 | . . . . . 6 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) | |
9 | 8 | exbii 1839 | . . . . 5 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) |
10 | 5, 7, 9 | 3bitr4ri 305 | . . . 4 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅) |
11 | rexnal 3235 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 = ∅) | |
12 | 4, 10, 11 | 3bitri 298 | . . 3 ⊢ (¬ ∪ 𝐴 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
13 | 12 | con4bii 322 | . 2 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
14 | 2, 3, 13 | 3bitr4ri 305 | 1 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ⊆ wss 3933 ∅c0 4288 {csn 4557 ∪ cuni 4830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-v 3494 df-dif 3936 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-uni 4831 |
This theorem is referenced by: uni0c 4856 uni0 4857 fin1a2lem11 9820 zornn0g 9915 0top 21519 filconn 22419 alexsubALTlem2 22584 ordcmp 33692 unisn0 41193 |
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