Theorem ordunisuc 7074

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 7030	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		suceq 5828	. . . . . 6 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
3	2	unieqd 4478	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ suc 𝑥 = ∪ suc 𝐴)
4		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
5	3, 4	eqeq12d 2666	. . . 4 ⊢ (𝑥 = 𝐴 → (∪ suc 𝑥 = 𝑥 ↔ ∪ suc 𝐴 = 𝐴))
6		eloni 5771	. . . . . 6 ⊢ (𝑥 ∈ On → Ord 𝑥)
7		ordtr 5775	. . . . . 6 ⊢ (Ord 𝑥 → Tr 𝑥)
8	6, 7	syl 17	. . . . 5 ⊢ (𝑥 ∈ On → Tr 𝑥)
9		vex 3234	. . . . . 6 ⊢ 𝑥 ∈ V
10	9	unisuc 5839	. . . . 5 ⊢ (Tr 𝑥 ↔ ∪ suc 𝑥 = 𝑥)
11	8, 10	sylib 208	. . . 4 ⊢ (𝑥 ∈ On → ∪ suc 𝑥 = 𝑥)
12	5, 11	vtoclga 3303	. . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
13		sucon 7050	. . . . . 6 ⊢ suc On = On
14	13	unieqi 4477	. . . . 5 ⊢ ∪ suc On = ∪ On
15		unon 7073	. . . . 5 ⊢ ∪ On = On
16	14, 15	eqtri 2673	. . . 4 ⊢ ∪ suc On = On
17		suceq 5828	. . . . 5 ⊢ (𝐴 = On → suc 𝐴 = suc On)
18	17	unieqd 4478	. . . 4 ⊢ (𝐴 = On → ∪ suc 𝐴 = ∪ suc On)
19		id 22	. . . 4 ⊢ (𝐴 = On → 𝐴 = On)
20	16, 18, 19	3eqtr4a 2711	. . 3 ⊢ (𝐴 = On → ∪ suc 𝐴 = 𝐴)
21	12, 20	jaoi 393	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → ∪ suc 𝐴 = 𝐴)
22	1, 21	sylbi 207	1 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)

Syntax hints:   → wi 4   ∨ wo 382   = wceq 1523   ∈ wcel 2030  ∪ cuni 4468  Tr wtr 4785  Ord word 5760  Oncon0 5761  suc csuc 5763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767

This theorem is referenced by:  orduniss2  7075  onsucuni2  7076  nlimsucg  7084  tz7.44-2  7548  ttukeylem7  9375  tsksuc  9622  dfrdg2  31825  ontgsucval  32556  onsuctopon  32558  limsucncmpi  32569  finxpsuclem  33364