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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 4866 | . 2 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅}) | |
2 | dfss3 3958 | . 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅}) | |
3 | velsn 4585 | . . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
4 | 3 | ralbii 3167 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
5 | 1, 2, 4 | 3bitri 299 | 1 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 ∅c0 4293 {csn 4569 ∪ cuni 4840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 df-sn 4570 df-uni 4841 |
This theorem is referenced by: fin1a2lem13 9836 fctop 21614 cctop 21616 ssmxidllem 30980 |
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