![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7721 | . 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
2 | suceq 6388 | . . . . . 6 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴) | |
3 | 2 | unieqd 4884 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ suc 𝑥 = ∪ suc 𝐴) |
4 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
5 | 3, 4 | eqeq12d 2753 | . . . 4 ⊢ (𝑥 = 𝐴 → (∪ suc 𝑥 = 𝑥 ↔ ∪ suc 𝐴 = 𝐴)) |
6 | eloni 6332 | . . . . . 6 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
7 | ordtr 6336 | . . . . . 6 ⊢ (Ord 𝑥 → Tr 𝑥) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ On → Tr 𝑥) |
9 | vex 3452 | . . . . . 6 ⊢ 𝑥 ∈ V | |
10 | 9 | unisuc 6401 | . . . . 5 ⊢ (Tr 𝑥 ↔ ∪ suc 𝑥 = 𝑥) |
11 | 8, 10 | sylib 217 | . . . 4 ⊢ (𝑥 ∈ On → ∪ suc 𝑥 = 𝑥) |
12 | 5, 11 | vtoclga 3537 | . . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) |
13 | sucon 7743 | . . . . . 6 ⊢ suc On = On | |
14 | 13 | unieqi 4883 | . . . . 5 ⊢ ∪ suc On = ∪ On |
15 | unon 7771 | . . . . 5 ⊢ ∪ On = On | |
16 | 14, 15 | eqtri 2765 | . . . 4 ⊢ ∪ suc On = On |
17 | suceq 6388 | . . . . 5 ⊢ (𝐴 = On → suc 𝐴 = suc On) | |
18 | 17 | unieqd 4884 | . . . 4 ⊢ (𝐴 = On → ∪ suc 𝐴 = ∪ suc On) |
19 | id 22 | . . . 4 ⊢ (𝐴 = On → 𝐴 = On) | |
20 | 16, 18, 19 | 3eqtr4a 2803 | . . 3 ⊢ (𝐴 = On → ∪ suc 𝐴 = 𝐴) |
21 | 12, 20 | jaoi 856 | . 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 846 = wceq 1542 ∈ wcel 2107 ∪ cuni 4870 Tr wtr 5227 Ord word 6321 Oncon0 6322 suc csuc 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-ord 6325 df-on 6326 df-suc 6328 |
This theorem is referenced by: orduniss2 7773 onsucuni2 7774 nlimsucg 7783 tz7.44-2 8358 rnttrcl 9665 ttrclselem2 9669 ttukeylem7 10458 tsksuc 10705 nnuni 34338 dfrdg2 34409 ontgsucval 34933 onsuctopon 34935 limsucncmpi 34946 finxpsuclem 35897 |
Copyright terms: Public domain | W3C validator |