Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4543 | . . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
2 | 1 | ralbii 3078 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
3 | dfss3 3875 | . 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅}) | |
4 | neq0 4246 | . . . 4 ⊢ (¬ ∪ 𝐴 = ∅ ↔ ∃𝑦 𝑦 ∈ ∪ 𝐴) | |
5 | rexcom4 3162 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) | |
6 | neq0 4246 | . . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥) | |
7 | 6 | rexbii 3160 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥) |
8 | eluni2 4809 | . . . . . 6 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) | |
9 | 8 | exbii 1855 | . . . . 5 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) |
10 | 5, 7, 9 | 3bitr4ri 307 | . . . 4 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅) |
11 | rexnal 3150 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 = ∅) | |
12 | 4, 10, 11 | 3bitri 300 | . . 3 ⊢ (¬ ∪ 𝐴 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
13 | 12 | con4bii 324 | . 2 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
14 | 2, 3, 13 | 3bitr4ri 307 | 1 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 ⊆ wss 3853 ∅c0 4223 {csn 4527 ∪ cuni 4805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-v 3400 df-dif 3856 df-in 3860 df-ss 3870 df-nul 4224 df-sn 4528 df-uni 4806 |
This theorem is referenced by: uni0c 4834 uni0 4835 fin1a2lem11 9989 zornn0g 10084 0top 21834 filconn 22734 alexsubALTlem2 22899 ordcmp 34322 unisn0 42216 |
Copyright terms: Public domain | W3C validator |