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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 3381 | . . . 4 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
2 | 1 | ralbii 2441 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 = ∅ ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥) |
3 | ralcom4 2708 | . . 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 3087 | . . 3 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅}) | |
6 | velsn 3544 | . . . 4 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
7 | 6 | ralbii 2441 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
8 | 5, 7 | bitri 183 | . 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
9 | eluni2 3740 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) | |
10 | 9 | notbii 657 | . . . 4 ⊢ (¬ 𝑦 ∈ ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) |
11 | 10 | albii 1446 | . . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ ∪ 𝐴 ↔ ∀𝑦 ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) |
12 | eq0 3381 | . . 3 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ ∪ 𝐴) | |
13 | ralnex 2426 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝑥 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) | |
14 | 13 | albii 1446 | . . 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 1329 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 ⊆ wss 3071 ∅c0 3363 {csn 3527 ∪ cuni 3736 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-nul 3364 df-sn 3533 df-uni 3737 |
This theorem is referenced by: uni0c 3762 uni0 3763 |
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