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Mirrors > Home > ILE Home > Th. List > 0iun | 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 3238 | . . . 4 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴 | |
2 | eliun 3661 | . . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴) | |
3 | 1, 2 | mtbir 596 | . . 3 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 |
4 | noel 3228 | . . 3 ⊢ ¬ 𝑦 ∈ ∅ | |
5 | 3, 4 | 2false 617 | . 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ 𝑦 ∈ ∅) |
6 | 5 | eqriv 2037 | 1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 ∃wrex 2307 ∅c0 3224 ∪ ciun 3657 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-nul 3225 df-iun 3659 |
This theorem is referenced by: iununir 3738 rdg0 5974 |
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