Theorem ordunisuc 7812

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 7765	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		suceq 6414	. . . . . 6 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
3	2	unieqd 4879	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ suc 𝑥 = ∪ suc 𝐴)
4		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
5	3, 4	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝐴 → (∪ suc 𝑥 = 𝑥 ↔ ∪ suc 𝐴 = 𝐴))
6		eloni 6356	. . . . . 6 ⊢ (𝑥 ∈ On → Ord 𝑥)
7		ordtr 6360	. . . . . 6 ⊢ (Ord 𝑥 → Tr 𝑥)
8	6, 7	syl 17	. . . . 5 ⊢ (𝑥 ∈ On → Tr 𝑥)
9		vex 3459	. . . . . 6 ⊢ 𝑥 ∈ V
10	9	unisuc 6427	. . . . 5 ⊢ (Tr 𝑥 ↔ ∪ suc 𝑥 = 𝑥)
11	8, 10	sylib 220	. . . 4 ⊢ (𝑥 ∈ On → ∪ suc 𝑥 = 𝑥)
12	5, 11	vtoclga 3542	. . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
13		sucon 7786	. . . . . 6 ⊢ suc On = On
14	13	unieqi 4878	. . . . 5 ⊢ ∪ suc On = ∪ On
15		unon 7811	. . . . 5 ⊢ ∪ On = On
16	14, 15	eqtri 2786	. . . 4 ⊢ ∪ suc On = On
17		suceq 6414	. . . . 5 ⊢ (𝐴 = On → suc 𝐴 = suc On)
18	17	unieqd 4879	. . . 4 ⊢ (𝐴 = On → ∪ suc 𝐴 = ∪ suc On)
19		id 22	. . . 4 ⊢ (𝐴 = On → 𝐴 = On)
20	16, 18, 19	3eqtr4a 2824	. . 3 ⊢ (𝐴 = On → ∪ suc 𝐴 = 𝐴)
21	12, 20	jaoi 868	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → ∪ suc 𝐴 = 𝐴)
22	1, 21	sylbi 219	1 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)

Syntax hints:   → wi 4   ∨ wo 858   = wceq 1561   ∈ wcel 2143  ∪ cuni 4866  Tr wtr 5208  Ord word 6345  Oncon0 6346  suc csuc 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-tr 5209  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-ord 6349  df-on 6350  df-suc 6352

This theorem is referenced by:  orduniss2  7813  onsucuni2  7814  nlimsucg  7822  tz7.44-2  8378  rnttrcl  9675  ttrclselem2  9679  ttukeylem7  10483  tsksuc  10731  nnuni  36082  dfrdg2  36148  ontgsucval  36797  onsuctopon  36799  limsucncmpi  36810  finxpsuclem  37896