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Mirrors > Home > MPE Home > Th. List > 0iun | Structured version Visualization version GIF version |
Description: An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
0iun | ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rex0 4360 | . . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴 | |
2 | eliun 5005 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴) | |
3 | 1, 2 | mtbir 322 | . 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 |
4 | 3 | nel0 4353 | 1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ∃wrex 3060 ∅c0 4325 ∪ ciun 5001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-v 3464 df-dif 3950 df-nul 4326 df-iun 5003 |
This theorem is referenced by: iinvdif 5088 iununi 5107 iunfi 9385 pwsdompw 10247 fsum2d 15775 fsumiun 15825 fprod2d 15983 prmreclem4 16921 prmreclem5 16922 fiuncmp 23399 ovolfiniun 25521 ovoliunnul 25527 finiunmbl 25564 volfiniun 25567 volsup 25576 gsumpart 32923 esum2dlem 33925 sigapildsyslem 33994 fiunelros 34007 mrsubvrs 35350 0totbnd 37474 totbndbnd 37490 fiiuncl 44666 sge0iunmptlemfi 46034 caragenfiiuncl 46136 carageniuncllem1 46142 |
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