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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 3407 | . . . 4 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴 | |
2 | eliun 3849 | . . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴) | |
3 | 1, 2 | mtbir 661 | . . 3 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 |
4 | noel 3394 | . . 3 ⊢ ¬ 𝑦 ∈ ∅ | |
5 | 3, 4 | 2false 691 | . 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ 𝑦 ∈ ∅) |
6 | 5 | eqriv 2151 | 1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 2125 ∃wrex 2433 ∅c0 3390 ∪ ciun 3845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-ext 2136 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ral 2437 df-rex 2438 df-v 2711 df-dif 3100 df-nul 3391 df-iun 3847 |
This theorem is referenced by: iununir 3928 rdg0 6324 iunfidisj 6879 fsum2d 11309 fsumiun 11351 fprod2d 11497 iuncld 12454 |
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