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Mirrors > Home > ILE Home > Th. List > uni0b | 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 | eq0 3386 | . . . 4 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
2 | 1 | ralbii 2444 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 = ∅ ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥) |
3 | ralcom4 2711 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝑥) | |
4 | 2, 3 | bitri 183 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 = ∅ ↔ ∀𝑦∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝑥) |
5 | dfss3 3092 | . . 3 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅}) | |
6 | velsn 3549 | . . . 4 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
7 | 6 | ralbii 2444 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
8 | 5, 7 | bitri 183 | . 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
9 | eluni2 3748 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) | |
10 | 9 | notbii 658 | . . . 4 ⊢ (¬ 𝑦 ∈ ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) |
11 | 10 | albii 1447 | . . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ ∪ 𝐴 ↔ ∀𝑦 ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) |
12 | eq0 3386 | . . 3 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ ∪ 𝐴) | |
13 | ralnex 2427 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝑥 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) | |
14 | 13 | albii 1447 | . . 3 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦 ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) |
15 | 11, 12, 14 | 3bitr4i 211 | . 2 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑦∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝑥) |
16 | 4, 8, 15 | 3bitr4ri 212 | 1 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅}) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 ∀wal 1330 = wceq 1332 ∈ wcel 1481 ∀wral 2417 ∃wrex 2418 ⊆ wss 3076 ∅c0 3368 {csn 3532 ∪ cuni 3744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-in 3082 df-ss 3089 df-nul 3369 df-sn 3538 df-uni 3745 |
This theorem is referenced by: uni0c 3770 uni0 3771 |
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