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Mirrors > Home > MPE Home > Th. List > ordunisuc | Structured version Visualization version GIF version |
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 7766 | . 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
2 | suceq 6424 | . . . . . 6 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴) | |
3 | 2 | unieqd 4915 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ suc 𝑥 = ∪ suc 𝐴) |
4 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
5 | 3, 4 | eqeq12d 2742 | . . . 4 ⊢ (𝑥 = 𝐴 → (∪ suc 𝑥 = 𝑥 ↔ ∪ suc 𝐴 = 𝐴)) |
6 | eloni 6368 | . . . . . 6 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
7 | ordtr 6372 | . . . . . 6 ⊢ (Ord 𝑥 → Tr 𝑥) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ On → Tr 𝑥) |
9 | vex 3472 | . . . . . 6 ⊢ 𝑥 ∈ V | |
10 | 9 | unisuc 6437 | . . . . 5 ⊢ (Tr 𝑥 ↔ ∪ suc 𝑥 = 𝑥) |
11 | 8, 10 | sylib 217 | . . . 4 ⊢ (𝑥 ∈ On → ∪ suc 𝑥 = 𝑥) |
12 | 5, 11 | vtoclga 3560 | . . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) |
13 | sucon 7788 | . . . . . 6 ⊢ suc On = On | |
14 | 13 | unieqi 4914 | . . . . 5 ⊢ ∪ suc On = ∪ On |
15 | unon 7816 | . . . . 5 ⊢ ∪ On = On | |
16 | 14, 15 | eqtri 2754 | . . . 4 ⊢ ∪ suc On = On |
17 | suceq 6424 | . . . . 5 ⊢ (𝐴 = On → suc 𝐴 = suc On) | |
18 | 17 | unieqd 4915 | . . . 4 ⊢ (𝐴 = On → ∪ suc 𝐴 = ∪ suc On) |
19 | id 22 | . . . 4 ⊢ (𝐴 = On → 𝐴 = On) | |
20 | 16, 18, 19 | 3eqtr4a 2792 | . . 3 ⊢ (𝐴 = On → ∪ suc 𝐴 = 𝐴) |
21 | 12, 20 | jaoi 854 | . 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → ∪ suc 𝐴 = 𝐴) |
22 | 1, 21 | sylbi 216 | 1 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ∪ cuni 4902 Tr wtr 5258 Ord word 6357 Oncon0 6358 suc csuc 6360 |
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 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-ord 6361 df-on 6362 df-suc 6364 |
This theorem is referenced by: orduniss2 7818 onsucuni2 7819 nlimsucg 7828 tz7.44-2 8408 rnttrcl 9719 ttrclselem2 9723 ttukeylem7 10512 tsksuc 10759 nnuni 35230 dfrdg2 35300 ontgsucval 35825 onsuctopon 35827 limsucncmpi 35838 finxpsuclem 36785 |
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