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Mirrors > Home > ILE Home > Th. List > ordsucim | GIF version |
Description: The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.) |
Ref | Expression |
---|---|
ordsucim | ⊢ (Ord 𝐴 → Ord suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtr 4229 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
2 | suctr 4272 | . . 3 ⊢ (Tr 𝐴 → Tr suc 𝐴) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (Ord 𝐴 → Tr suc 𝐴) |
4 | df-suc 4222 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | 4 | eleq2i 2161 | . . . . 5 ⊢ (𝑥 ∈ suc 𝐴 ↔ 𝑥 ∈ (𝐴 ∪ {𝐴})) |
6 | elun 3156 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴})) | |
7 | velsn 3483 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
8 | 7 | orbi2i 717 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴)) |
9 | 5, 6, 8 | 3bitri 205 | . . . 4 ⊢ (𝑥 ∈ suc 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴)) |
10 | dford3 4218 | . . . . . . . 8 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) | |
11 | 10 | simprbi 270 | . . . . . . 7 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 Tr 𝑥) |
12 | df-ral 2375 | . . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → Tr 𝑥)) | |
13 | 11, 12 | sylib 121 | . . . . . 6 ⊢ (Ord 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → Tr 𝑥)) |
14 | 13 | 19.21bi 1502 | . . . . 5 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → Tr 𝑥)) |
15 | treq 3964 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (Tr 𝑥 ↔ Tr 𝐴)) | |
16 | 1, 15 | syl5ibrcom 156 | . . . . 5 ⊢ (Ord 𝐴 → (𝑥 = 𝐴 → Tr 𝑥)) |
17 | 14, 16 | jaod 675 | . . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴) → Tr 𝑥)) |
18 | 9, 17 | syl5bi 151 | . . 3 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → Tr 𝑥)) |
19 | 18 | ralrimiv 2457 | . 2 ⊢ (Ord 𝐴 → ∀𝑥 ∈ suc 𝐴Tr 𝑥) |
20 | dford3 4218 | . 2 ⊢ (Ord suc 𝐴 ↔ (Tr suc 𝐴 ∧ ∀𝑥 ∈ suc 𝐴Tr 𝑥)) | |
21 | 3, 19, 20 | sylanbrc 409 | 1 ⊢ (Ord 𝐴 → Ord suc 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 667 ∀wal 1294 = wceq 1296 ∈ wcel 1445 ∀wral 2370 ∪ cun 3011 {csn 3466 Tr wtr 3958 Ord word 4213 suc csuc 4216 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-sn 3472 df-uni 3676 df-tr 3959 df-iord 4217 df-suc 4222 |
This theorem is referenced by: suceloni 4346 ordsucg 4347 onsucsssucr 4354 ordtriexmidlem 4364 2ordpr 4368 ordsuc 4407 nnsucsssuc 6293 |
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