![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iun0 | Structured version Visualization version GIF version |
Description: An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
iun0 | ⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4343 | . . . . 5 ⊢ ¬ 𝑦 ∈ ∅ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑦 ∈ ∅) |
3 | 2 | nrex 3071 | . . 3 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅ |
4 | eliun 4999 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅) | |
5 | 3, 4 | mtbir 323 | . 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ |
6 | 5 | nel0 4359 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 ∅c0 4338 ∪ ciun 4995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-v 3479 df-dif 3965 df-nul 4339 df-iun 4997 |
This theorem is referenced by: iunxdif3 5099 iununi 5103 funiunfv 7267 om0r 8575 kmlem11 10198 ituniiun 10459 dfrtrclrec2 15093 voliunlem1 25598 ofpreima2 32682 ssdifidllem 33463 esum2dlem 34072 sigaclfu2 34101 measvunilem0 34193 measvuni 34194 cvmscld 35257 ovolval4lem1 46604 |
Copyright terms: Public domain | W3C validator |