Theorem ordunisuc 7833

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.)

Assertion

Ref	Expression
ordunisuc	⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)

Proof of Theorem ordunisuc

Dummy variable 𝑥 is distinct from all other variables.

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)

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