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| Description: An ordinal class is equal to the union of its successor. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) | 
| Ref | Expression | 
|---|---|
| ordunisuc | ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ordeleqon 7803 | . 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
| 2 | suceq 6449 | . . . . . 6 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴) | |
| 3 | 2 | unieqd 4919 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ suc 𝑥 = ∪ suc 𝐴) | 
| 4 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
| 5 | 3, 4 | eqeq12d 2752 | . . . 4 ⊢ (𝑥 = 𝐴 → (∪ suc 𝑥 = 𝑥 ↔ ∪ suc 𝐴 = 𝐴)) | 
| 6 | eloni 6393 | . . . . . 6 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
| 7 | ordtr 6397 | . . . . . 6 ⊢ (Ord 𝑥 → Tr 𝑥) | |
| 8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ On → Tr 𝑥) | 
| 9 | vex 3483 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 10 | 9 | unisuc 6462 | . . . . 5 ⊢ (Tr 𝑥 ↔ ∪ suc 𝑥 = 𝑥) | 
| 11 | 8, 10 | sylib 218 | . . . 4 ⊢ (𝑥 ∈ On → ∪ suc 𝑥 = 𝑥) | 
| 12 | 5, 11 | vtoclga 3576 | . . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) | 
| 13 | sucon 7824 | . . . . . 6 ⊢ suc On = On | |
| 14 | 13 | unieqi 4918 | . . . . 5 ⊢ ∪ suc On = ∪ On | 
| 15 | unon 7852 | . . . . 5 ⊢ ∪ On = On | |
| 16 | 14, 15 | eqtri 2764 | . . . 4 ⊢ ∪ suc On = On | 
| 17 | suceq 6449 | . . . . 5 ⊢ (𝐴 = On → suc 𝐴 = suc On) | |
| 18 | 17 | unieqd 4919 | . . . 4 ⊢ (𝐴 = On → ∪ suc 𝐴 = ∪ suc On) | 
| 19 | id 22 | . . . 4 ⊢ (𝐴 = On → 𝐴 = On) | |
| 20 | 16, 18, 19 | 3eqtr4a 2802 | . . 3 ⊢ (𝐴 = On → ∪ suc 𝐴 = 𝐴) | 
| 21 | 12, 20 | jaoi 857 | . 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → ∪ suc 𝐴 = 𝐴) | 
| 22 | 1, 21 | sylbi 217 | 1 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ∪ cuni 4906 Tr wtr 5258 Ord word 6382 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 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 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: orduniss2 7854 onsucuni2 7855 nlimsucg 7864 tz7.44-2 8448 rnttrcl 9763 ttrclselem2 9767 ttukeylem7 10556 tsksuc 10803 nnuni 35728 dfrdg2 35797 ontgsucval 36434 onsuctopon 36436 limsucncmpi 36447 finxpsuclem 37399 | 
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