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Mirrors > Home > ILE Home > Th. List > uni0c | GIF version |
Description: The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.) |
Ref | Expression |
---|---|
uni0c | ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uni0b 3731 | . 2 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅}) | |
2 | dfss3 3057 | . 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅}) | |
3 | velsn 3514 | . . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
4 | 3 | ralbii 2418 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
5 | 1, 2, 4 | 3bitri 205 | 1 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 ∈ wcel 1465 ∀wral 2393 ⊆ wss 3041 ∅c0 3333 {csn 3497 ∪ cuni 3706 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-in 3047 df-ss 3054 df-nul 3334 df-sn 3503 df-uni 3707 |
This theorem is referenced by: (None) |
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