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| Mirrors > Home > MPE Home > Th. List > iunsuc | Structured version Visualization version 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 6390 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
| 2 | iuneq1 5008 | . . 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 5094 | . 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 5091 | . . 3 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
| 8 | 7 | uneq2i 4165 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝐴}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) |
| 9 | 3, 4, 8 | 3eqtri 2769 | 1 ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {csn 4626 ∪ ciun 4991 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-iun 4993 df-suc 6390 |
| This theorem is referenced by: pwsdompw 10243 |
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