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Mirrors > Home > MPE Home > Th. List > uni0c | Structured version Visualization version 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 4826 | . 2 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅}) | |
2 | dfss3 3903 | . 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅}) | |
3 | velsn 4541 | . . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
4 | 3 | ralbii 3133 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
5 | 1, 2, 4 | 3bitri 300 | 1 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 ∅c0 4243 {csn 4525 ∪ cuni 4800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-11 2158 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-uni 4801 |
This theorem is referenced by: fin1a2lem13 9823 fctop 21609 cctop 21611 ssmxidllem 31049 |
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