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Mirrors > Home > MPE Home > Th. List > oneluni | Structured version Visualization version GIF version |
Description: An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.) |
Ref | Expression |
---|---|
on.1 | ⊢ 𝐴 ∈ On |
Ref | Expression |
---|---|
oneluni | ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | on.1 | . . 3 ⊢ 𝐴 ∈ On | |
2 | 1 | onelssi 6472 | . 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴) |
3 | ssequn2 4178 | . 2 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴) | |
4 | 2, 3 | sylib 217 | 1 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3941 ⊆ wss 3943 Oncon0 6357 |
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-ral 3056 df-v 3470 df-un 3948 df-in 3950 df-ss 3960 df-uni 4903 df-tr 5259 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-ord 6360 df-on 6361 |
This theorem is referenced by: omabs2 42640 |
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