Theorem ordsucim 4621

Description: The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)

Assertion

Ref	Expression
ordsucim	⊢ (Ord 𝐴 → Ord suc 𝐴)

Proof of Theorem ordsucim

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtr 4498	. . 3 ⊢ (Ord 𝐴 → Tr 𝐴)
2		suctr 4541	. . 3 ⊢ (Tr 𝐴 → Tr suc 𝐴)
3	1, 2	syl 14	. 2 ⊢ (Ord 𝐴 → Tr suc 𝐴)
4		df-suc 4491	. . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5	4	eleq2i 2299	. . . . 5 ⊢ (𝑥 ∈ suc 𝐴 ↔ 𝑥 ∈ (𝐴 ∪ {𝐴}))
6		elun 3359	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
7		velsn 3705	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8	7	orbi2i 770	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴))
9	5, 6, 8	3bitri 206	. . . 4 ⊢ (𝑥 ∈ suc 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴))
10		dford3 4487	. . . . . . . 8 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
11	10	simprbi 275	. . . . . . 7 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 Tr 𝑥)
12		df-ral 2525	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → Tr 𝑥))
13	11, 12	sylib 122	. . . . . 6 ⊢ (Ord 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → Tr 𝑥))
14	13	19.21bi 1607	. . . . 5 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → Tr 𝑥))
15		treq 4213	. . . . . 6 ⊢ (𝑥 = 𝐴 → (Tr 𝑥 ↔ Tr 𝐴))
16	1, 15	syl5ibrcom 157	. . . . 5 ⊢ (Ord 𝐴 → (𝑥 = 𝐴 → Tr 𝑥))
17	14, 16	jaod 725	. . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴) → Tr 𝑥))
18	9, 17	biimtrid 152	. . 3 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → Tr 𝑥))
19	18	ralrimiv 2614	. 2 ⊢ (Ord 𝐴 → ∀𝑥 ∈ suc 𝐴Tr 𝑥)
20		dford3 4487	. 2 ⊢ (Ord suc 𝐴 ↔ (Tr suc 𝐴 ∧ ∀𝑥 ∈ suc 𝐴Tr 𝑥))
21	3, 19, 20	sylanbrc 417	1 ⊢ (Ord 𝐴 → Ord suc 𝐴)

Syntax hints:   → wi 4   ∨ wo 716  ∀wal 1396   = wceq 1398   ∈ wcel 2203  ∀wral 2520   ∪ cun 3208  {csn 3688  Tr wtr 4207  Ord word 4482  suc csuc 4485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-sn 3694  df-uni 3914  df-tr 4208  df-iord 4486  df-suc 4491

This theorem is referenced by:  onsuc  4622  ordsucg  4623  onsucsssucr  4630  ordtriexmidlem  4640  2ordpr  4645  ordsuc  4684  nnsucsssuc  6724