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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 4115 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
2 | suctr 4158 | . . 3 ⊢ (Tr 𝐴 → Tr suc 𝐴) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (Ord 𝐴 → Tr suc 𝐴) |
4 | df-suc 4108 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | 4 | eleq2i 2104 | . . . . 5 ⊢ (𝑥 ∈ suc 𝐴 ↔ 𝑥 ∈ (𝐴 ∪ {𝐴})) |
6 | elun 3084 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴})) | |
7 | velsn 3392 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
8 | 7 | orbi2i 679 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴)) |
9 | 5, 6, 8 | 3bitri 195 | . . . 4 ⊢ (𝑥 ∈ suc 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴)) |
10 | dford3 4104 | . . . . . . . 8 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) | |
11 | 10 | simprbi 260 | . . . . . . 7 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 Tr 𝑥) |
12 | df-ral 2311 | . . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → Tr 𝑥)) | |
13 | 11, 12 | sylib 127 | . . . . . 6 ⊢ (Ord 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → Tr 𝑥)) |
14 | 13 | 19.21bi 1450 | . . . . 5 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → Tr 𝑥)) |
15 | treq 3860 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (Tr 𝑥 ↔ Tr 𝐴)) | |
16 | 1, 15 | syl5ibrcom 146 | . . . . 5 ⊢ (Ord 𝐴 → (𝑥 = 𝐴 → Tr 𝑥)) |
17 | 14, 16 | jaod 637 | . . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴) → Tr 𝑥)) |
18 | 9, 17 | syl5bi 141 | . . 3 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → Tr 𝑥)) |
19 | 18 | ralrimiv 2391 | . 2 ⊢ (Ord 𝐴 → ∀𝑥 ∈ suc 𝐴Tr 𝑥) |
20 | dford3 4104 | . 2 ⊢ (Ord suc 𝐴 ↔ (Tr suc 𝐴 ∧ ∀𝑥 ∈ suc 𝐴Tr 𝑥)) | |
21 | 3, 19, 20 | sylanbrc 394 | 1 ⊢ (Ord 𝐴 → Ord suc 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 629 ∀wal 1241 = wceq 1243 ∈ wcel 1393 ∀wral 2306 ∪ cun 2915 {csn 3375 Tr wtr 3854 Ord word 4099 suc csuc 4102 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-uni 3581 df-tr 3855 df-iord 4103 df-suc 4108 |
This theorem is referenced by: suceloni 4227 ordsucg 4228 onsucsssucr 4235 ordtriexmidlem 4245 2ordpr 4249 ordsuc 4287 nnsucsssuc 6071 |
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