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Mirrors > Home > ILE Home > Th. List > iunsuc | GIF version |
Description: Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
iunsuc.1 | ⊢ 𝐴 ∈ V |
iunsuc.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iunsuc | ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 4293 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | iuneq1 3826 | . . 3 ⊢ (suc 𝐴 = (𝐴 ∪ {𝐴}) → ∪ 𝑥 ∈ suc 𝐴𝐵 = ∪ 𝑥 ∈ (𝐴 ∪ {𝐴})𝐵) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = ∪ 𝑥 ∈ (𝐴 ∪ {𝐴})𝐵 |
4 | iunxun 3892 | . 2 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ {𝐴})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝐴}𝐵) | |
5 | iunsuc.1 | . . . 4 ⊢ 𝐴 ∈ V | |
6 | iunsuc.2 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
7 | 5, 6 | iunxsn 3889 | . . 3 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
8 | 7 | uneq2i 3227 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝐴}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) |
9 | 3, 4, 8 | 3eqtri 2164 | 1 ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 {csn 3527 ∪ ciun 3813 suc csuc 4287 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-iun 3815 df-suc 4293 |
This theorem is referenced by: (None) |
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