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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 4301 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | iuneq1 3834 | . . 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 3900 | . 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 3897 | . . 3 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
8 | 7 | uneq2i 3232 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝐴}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) |
9 | 3, 4, 8 | 3eqtri 2165 | 1 ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ∪ cun 3074 {csn 3532 ∪ ciun 3821 suc csuc 4295 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-iun 3823 df-suc 4301 |
This theorem is referenced by: (None) |
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