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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 7765 | . 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
2 | suceq 6427 | . . . . . 6 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴) | |
3 | 2 | unieqd 4921 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ suc 𝑥 = ∪ suc 𝐴) |
4 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
5 | 3, 4 | eqeq12d 2748 | . . . 4 ⊢ (𝑥 = 𝐴 → (∪ suc 𝑥 = 𝑥 ↔ ∪ suc 𝐴 = 𝐴)) |
6 | eloni 6371 | . . . . . 6 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
7 | ordtr 6375 | . . . . . 6 ⊢ (Ord 𝑥 → Tr 𝑥) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ On → Tr 𝑥) |
9 | vex 3478 | . . . . . 6 ⊢ 𝑥 ∈ V | |
10 | 9 | unisuc 6440 | . . . . 5 ⊢ (Tr 𝑥 ↔ ∪ suc 𝑥 = 𝑥) |
11 | 8, 10 | sylib 217 | . . . 4 ⊢ (𝑥 ∈ On → ∪ suc 𝑥 = 𝑥) |
12 | 5, 11 | vtoclga 3565 | . . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) |
13 | sucon 7787 | . . . . . 6 ⊢ suc On = On | |
14 | 13 | unieqi 4920 | . . . . 5 ⊢ ∪ suc On = ∪ On |
15 | unon 7815 | . . . . 5 ⊢ ∪ On = On | |
16 | 14, 15 | eqtri 2760 | . . . 4 ⊢ ∪ suc On = On |
17 | suceq 6427 | . . . . 5 ⊢ (𝐴 = On → suc 𝐴 = suc On) | |
18 | 17 | unieqd 4921 | . . . 4 ⊢ (𝐴 = On → ∪ suc 𝐴 = ∪ suc On) |
19 | id 22 | . . . 4 ⊢ (𝐴 = On → 𝐴 = On) | |
20 | 16, 18, 19 | 3eqtr4a 2798 | . . 3 ⊢ (𝐴 = On → ∪ suc 𝐴 = 𝐴) |
21 | 12, 20 | jaoi 855 | . 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 845 = wceq 1541 ∈ wcel 2106 ∪ cuni 4907 Tr wtr 5264 Ord word 6360 Oncon0 6361 suc csuc 6363 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-ord 6364 df-on 6365 df-suc 6367 |
This theorem is referenced by: orduniss2 7817 onsucuni2 7818 nlimsucg 7827 tz7.44-2 8403 rnttrcl 9713 ttrclselem2 9717 ttukeylem7 10506 tsksuc 10753 nnuni 34684 dfrdg2 34755 ontgsucval 35305 onsuctopon 35307 limsucncmpi 35318 finxpsuclem 36266 |
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