![]() |
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 7782 | . 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
2 | suceq 6430 | . . . . . 6 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴) | |
3 | 2 | unieqd 4916 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ suc 𝑥 = ∪ suc 𝐴) |
4 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
5 | 3, 4 | eqeq12d 2741 | . . . 4 ⊢ (𝑥 = 𝐴 → (∪ suc 𝑥 = 𝑥 ↔ ∪ suc 𝐴 = 𝐴)) |
6 | eloni 6374 | . . . . . 6 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
7 | ordtr 6378 | . . . . . 6 ⊢ (Ord 𝑥 → Tr 𝑥) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ On → Tr 𝑥) |
9 | vex 3467 | . . . . . 6 ⊢ 𝑥 ∈ V | |
10 | 9 | unisuc 6443 | . . . . 5 ⊢ (Tr 𝑥 ↔ ∪ suc 𝑥 = 𝑥) |
11 | 8, 10 | sylib 217 | . . . 4 ⊢ (𝑥 ∈ On → ∪ suc 𝑥 = 𝑥) |
12 | 5, 11 | vtoclga 3555 | . . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) |
13 | sucon 7804 | . . . . . 6 ⊢ suc On = On | |
14 | 13 | unieqi 4915 | . . . . 5 ⊢ ∪ suc On = ∪ On |
15 | unon 7832 | . . . . 5 ⊢ ∪ On = On | |
16 | 14, 15 | eqtri 2753 | . . . 4 ⊢ ∪ suc On = On |
17 | suceq 6430 | . . . . 5 ⊢ (𝐴 = On → suc 𝐴 = suc On) | |
18 | 17 | unieqd 4916 | . . . 4 ⊢ (𝐴 = On → ∪ suc 𝐴 = ∪ suc On) |
19 | id 22 | . . . 4 ⊢ (𝐴 = On → 𝐴 = On) | |
20 | 16, 18, 19 | 3eqtr4a 2791 | . . 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 1533 ∈ wcel 2098 ∪ cuni 4903 Tr wtr 5260 Ord word 6363 Oncon0 6364 suc csuc 6366 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-tr 5261 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6367 df-on 6368 df-suc 6370 |
This theorem is referenced by: orduniss2 7834 onsucuni2 7835 nlimsucg 7844 tz7.44-2 8426 rnttrcl 9745 ttrclselem2 9749 ttukeylem7 10538 tsksuc 10785 nnuni 35378 dfrdg2 35448 ontgsucval 35973 onsuctopon 35975 limsucncmpi 35986 finxpsuclem 36933 |
Copyright terms: Public domain | W3C validator |