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Mirrors > Home > ILE Home > Th. List > iunxsn | GIF version |
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.) |
Ref | Expression |
---|---|
iunxsn.1 | ⊢ 𝐴 ∈ V |
iunxsn.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iunxsn | ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | iunxsn.2 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
3 | 2 | iunxsng 3883 | . 2 ⊢ (𝐴 ∈ V → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2681 {csn 3522 ∪ ciun 3808 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-sn 3528 df-iun 3810 |
This theorem is referenced by: iunsuc 4337 fsum2dlemstep 11196 fsumiun 11239 |
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