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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 7801 | . 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
2 | suceq 6452 | . . . . . 6 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴) | |
3 | 2 | unieqd 4925 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ suc 𝑥 = ∪ suc 𝐴) |
4 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
5 | 3, 4 | eqeq12d 2751 | . . . 4 ⊢ (𝑥 = 𝐴 → (∪ suc 𝑥 = 𝑥 ↔ ∪ suc 𝐴 = 𝐴)) |
6 | eloni 6396 | . . . . . 6 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
7 | ordtr 6400 | . . . . . 6 ⊢ (Ord 𝑥 → Tr 𝑥) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ On → Tr 𝑥) |
9 | vex 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
10 | 9 | unisuc 6465 | . . . . 5 ⊢ (Tr 𝑥 ↔ ∪ suc 𝑥 = 𝑥) |
11 | 8, 10 | sylib 218 | . . . 4 ⊢ (𝑥 ∈ On → ∪ suc 𝑥 = 𝑥) |
12 | 5, 11 | vtoclga 3577 | . . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) |
13 | sucon 7823 | . . . . . 6 ⊢ suc On = On | |
14 | 13 | unieqi 4924 | . . . . 5 ⊢ ∪ suc On = ∪ On |
15 | unon 7851 | . . . . 5 ⊢ ∪ On = On | |
16 | 14, 15 | eqtri 2763 | . . . 4 ⊢ ∪ suc On = On |
17 | suceq 6452 | . . . . 5 ⊢ (𝐴 = On → suc 𝐴 = suc On) | |
18 | 17 | unieqd 4925 | . . . 4 ⊢ (𝐴 = On → ∪ suc 𝐴 = ∪ suc On) |
19 | id 22 | . . . 4 ⊢ (𝐴 = On → 𝐴 = On) | |
20 | 16, 18, 19 | 3eqtr4a 2801 | . . 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 1537 ∈ wcel 2106 ∪ cuni 4912 Tr wtr 5265 Ord word 6385 Oncon0 6386 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-suc 6392 |
This theorem is referenced by: orduniss2 7853 onsucuni2 7854 nlimsucg 7863 tz7.44-2 8446 rnttrcl 9760 ttrclselem2 9764 ttukeylem7 10553 tsksuc 10800 nnuni 35707 dfrdg2 35777 ontgsucval 36415 onsuctopon 36417 limsucncmpi 36428 finxpsuclem 37380 |
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