Theorem ordunisuc 7827

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 7780	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		suceq 6429	. . . . . 6 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
3	2	unieqd 4884	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ suc 𝑥 = ∪ suc 𝐴)
4		id 23	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
5	3, 4	eqeq12d 2777	. . . 4 ⊢ (𝑥 = 𝐴 → (∪ suc 𝑥 = 𝑥 ↔ ∪ suc 𝐴 = 𝐴))
6		eloni 6370	. . . . . 6 ⊢ (𝑥 ∈ On → Ord 𝑥)
7		ordtr 6374	. . . . . 6 ⊢ (Ord 𝑥 → Tr 𝑥)
8	6, 7	syl 18	. . . . 5 ⊢ (𝑥 ∈ On → Tr 𝑥)
9		vex 3457	. . . . . 6 ⊢ 𝑥 ∈ V
10	9	unisuc 6442	. . . . 5 ⊢ (Tr 𝑥 ↔ ∪ suc 𝑥 = 𝑥)
11	8, 10	sylib 221	. . . 4 ⊢ (𝑥 ∈ On → ∪ suc 𝑥 = 𝑥)
12	5, 11	vtoclga 3540	. . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
13		sucon 7801	. . . . . 6 ⊢ suc On = On
14	13	unieqi 4883	. . . . 5 ⊢ ∪ suc On = ∪ On
15		unon 7826	. . . . 5 ⊢ ∪ On = On
16	14, 15	eqtri 2784	. . . 4 ⊢ ∪ suc On = On
17		suceq 6429	. . . . 5 ⊢ (𝐴 = On → suc 𝐴 = suc On)
18	17	unieqd 4884	. . . 4 ⊢ (𝐴 = On → ∪ suc 𝐴 = ∪ suc On)
19		id 23	. . . 4 ⊢ (𝐴 = On → 𝐴 = On)
20	16, 18, 19	3eqtr4a 2822	. . 3 ⊢ (𝐴 = On → ∪ suc 𝐴 = 𝐴)
21	12, 20	jaoi 870	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → ∪ suc 𝐴 = 𝐴)
22	1, 21	sylbi 220	1 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1568   ∈ wcel 2141  ∪ cuni 4871  Tr wtr 5217  Ord word 6359  Oncon0 6360  suc csuc 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-tr 5218  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364  df-suc 6366

This theorem is referenced by:  orduniss2  7828  onsucuni2  7829  nlimsucg  7837  tz7.44-2  8393  rnttrcl  9690  ttrclselem2  9694  ttukeylem7  10498  tsksuc  10746  nnuni  36185  dfrdg2  36251  ontgsucval  36909  onsuctopon  36911  limsucncmpi  36922  finxpsuclem  38009