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Mirrors > Home > MPE Home > Th. List > unisucs | Structured version Visualization version GIF version |
Description: The union of the successor of a set is equal to the binary union of that set with its union. (Contributed by NM, 30-Aug-1993.) Extract from unisuc 6436. (Revised by BJ, 28-Dec-2024.) |
Ref | Expression |
---|---|
unisucs | ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 6363 | . . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | unieqi 4914 | . . 3 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴}) |
3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})) |
4 | uniun 4927 | . . 3 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴}) | |
5 | 4 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})) |
6 | unisng 4922 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | |
7 | 6 | uneq2d 4158 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴)) |
8 | 3, 5, 7 | 3eqtrd 2770 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3941 {csn 4623 ∪ cuni 4902 suc csuc 6359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-un 3948 df-in 3950 df-ss 3960 df-sn 4624 df-pr 4626 df-uni 4903 df-suc 6363 |
This theorem is referenced by: unisucg 6435 |
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