Theorem ordunisuc 7831

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 7781	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		suceq 6424	. . . . . 6 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
3	2	unieqd 4901	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ suc 𝑥 = ∪ suc 𝐴)
4		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
5	3, 4	eqeq12d 2752	. . . 4 ⊢ (𝑥 = 𝐴 → (∪ suc 𝑥 = 𝑥 ↔ ∪ suc 𝐴 = 𝐴))
6		eloni 6367	. . . . . 6 ⊢ (𝑥 ∈ On → Ord 𝑥)
7		ordtr 6371	. . . . . 6 ⊢ (Ord 𝑥 → Tr 𝑥)
8	6, 7	syl 17	. . . . 5 ⊢ (𝑥 ∈ On → Tr 𝑥)
9		vex 3468	. . . . . 6 ⊢ 𝑥 ∈ V
10	9	unisuc 6438	. . . . 5 ⊢ (Tr 𝑥 ↔ ∪ suc 𝑥 = 𝑥)
11	8, 10	sylib 218	. . . 4 ⊢ (𝑥 ∈ On → ∪ suc 𝑥 = 𝑥)
12	5, 11	vtoclga 3561	. . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
13		sucon 7802	. . . . . 6 ⊢ suc On = On
14	13	unieqi 4900	. . . . 5 ⊢ ∪ suc On = ∪ On
15		unon 7830	. . . . 5 ⊢ ∪ On = On
16	14, 15	eqtri 2759	. . . 4 ⊢ ∪ suc On = On
17		suceq 6424	. . . . 5 ⊢ (𝐴 = On → suc 𝐴 = suc On)
18	17	unieqd 4901	. . . 4 ⊢ (𝐴 = On → ∪ suc 𝐴 = ∪ suc On)
19		id 22	. . . 4 ⊢ (𝐴 = On → 𝐴 = On)
20	16, 18, 19	3eqtr4a 2797	. . 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)

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∪ cuni 4888  Tr wtr 5234  Ord word 6356  Oncon0 6357  suc csuc 6359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6360  df-on 6361  df-suc 6363

This theorem is referenced by:  orduniss2  7832  onsucuni2  7833  nlimsucg  7842  tz7.44-2  8426  rnttrcl  9741  ttrclselem2  9745  ttukeylem7  10534  tsksuc  10781  nnuni  35749  dfrdg2  35818  ontgsucval  36455  onsuctopon  36457  limsucncmpi  36468  finxpsuclem  37420