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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 6430 | . 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴) |
3 | ssequn2 4142 | . 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 1541 ∈ wcel 2106 ∪ cun 3907 ⊆ wss 3909 Oncon0 6316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3064 df-v 3446 df-un 3914 df-in 3916 df-ss 3926 df-uni 4865 df-tr 5222 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-ord 6319 df-on 6320 |
This theorem is referenced by: omabs2 41641 |
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