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| Mirrors > Home > MPE Home > Th. List > onunisuc | Structured version Visualization version GIF version | ||
| Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6503. (Revised by BJ, 28-Dec-2024.) | 
| Ref | Expression | 
|---|---|
| onunisuc | ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ontr 6492 | . 2 ⊢ (𝐴 ∈ On → Tr 𝐴) | |
| 2 | unisucg 6461 | . 2 ⊢ (𝐴 ∈ On → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)) | |
| 3 | 1, 2 | mpbid 232 | 1 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∪ cuni 4906 Tr wtr 5258 Oncon0 6383 suc csuc 6385 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-v 3481 df-un 3955 df-ss 3967 df-sn 4626 df-pr 4628 df-uni 4907 df-tr 5259 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 df-suc 6389 | 
| This theorem is referenced by: onunisuci 6503 nlimsuc 43459 | 
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