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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 3878 | . . . . . 6 ⊢ ¬ 𝑦 ∈ ∅ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑦 ∈ ∅) |
3 | 2 | nrex 2983 | . . . 4 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅ |
4 | eliun 4460 | . . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅) | |
5 | 3, 4 | mtbir 312 | . . 3 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ |
6 | 5, 1 | 2false 364 | . 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ 𝑦 ∈ ∅) |
7 | 6 | eqriv 2607 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∅c0 3874 ∪ ciun 4455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-nul 3875 df-iun 4457 |
This theorem is referenced by: iunxdif3 4542 iununi 4546 funiunfv 6410 om0r 7506 kmlem11 8865 ituniiun 9127 voliunlem1 23125 ofpreima2 28849 esum2dlem 29481 sigaclfu2 29511 measvunilem0 29603 measvuni 29604 cvmscld 30509 trpred0 30980 ovolval4lem1 39539 |
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