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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 4331 | . . . . 5 ⊢ ¬ 𝑦 ∈ ∅ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑦 ∈ ∅) |
3 | 2 | nrex 3075 | . . 3 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅ |
4 | eliun 5002 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅) | |
5 | 3, 4 | mtbir 323 | . 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ |
6 | 5 | nel0 4351 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 ∅c0 4323 ∪ ciun 4998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-v 3477 df-dif 3952 df-nul 4324 df-iun 5000 |
This theorem is referenced by: iunxdif3 5099 iununi 5103 funiunfv 7247 om0r 8539 kmlem11 10155 ituniiun 10417 dfrtrclrec2 15005 voliunlem1 25067 ofpreima2 31891 esum2dlem 33090 sigaclfu2 33119 measvunilem0 33211 measvuni 33212 cvmscld 34264 ovolval4lem1 45365 |
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