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 4277 | . . . . 5 ⊢ ¬ 𝑦 ∈ ∅ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑦 ∈ ∅) |
3 | 2 | nrex 3074 | . . 3 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅ |
4 | eliun 4945 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅) | |
5 | 3, 4 | mtbir 322 | . 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ |
6 | 5 | nel0 4297 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2105 ∃wrex 3070 ∅c0 4269 ∪ ciun 4941 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3062 df-rex 3071 df-v 3443 df-dif 3901 df-nul 4270 df-iun 4943 |
This theorem is referenced by: iunxdif3 5042 iununi 5046 funiunfv 7177 om0r 8440 kmlem11 10017 ituniiun 10279 dfrtrclrec2 14868 voliunlem1 24820 ofpreima2 31290 esum2dlem 32358 sigaclfu2 32387 measvunilem0 32479 measvuni 32480 cvmscld 33534 ovolval4lem1 44524 |
Copyright terms: Public domain | W3C validator |