Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4338 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
2 | suctr 4381 | . . 3 ⊢ (Tr 𝐴 → Tr suc 𝐴) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (Ord 𝐴 → Tr suc 𝐴) |
4 | df-suc 4331 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | 4 | eleq2i 2224 | . . . . 5 ⊢ (𝑥 ∈ suc 𝐴 ↔ 𝑥 ∈ (𝐴 ∪ {𝐴})) |
6 | elun 3248 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴})) | |
7 | velsn 3577 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
8 | 7 | orbi2i 752 | . . . . 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 4327 | . . . . . . . 8 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) | |
11 | 10 | simprbi 273 | . . . . . . 7 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 Tr 𝑥) |
12 | df-ral 2440 | . . . . . . 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 1538 | . . . . 5 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → Tr 𝑥)) |
15 | treq 4068 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (Tr 𝑥 ↔ Tr 𝐴)) | |
16 | 1, 15 | syl5ibrcom 156 | . . . . 5 ⊢ (Ord 𝐴 → (𝑥 = 𝐴 → Tr 𝑥)) |
17 | 14, 16 | jaod 707 | . . . 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 2529 | . 2 ⊢ (Ord 𝐴 → ∀𝑥 ∈ suc 𝐴Tr 𝑥) |
20 | dford3 4327 | . 2 ⊢ (Ord suc 𝐴 ↔ (Tr suc 𝐴 ∧ ∀𝑥 ∈ suc 𝐴Tr 𝑥)) | |
21 | 3, 19, 20 | sylanbrc 414 | 1 ⊢ (Ord 𝐴 → Ord suc 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 ∀wal 1333 = wceq 1335 ∈ wcel 2128 ∀wral 2435 ∪ cun 3100 {csn 3560 Tr wtr 4062 Ord word 4322 suc csuc 4325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-uni 3773 df-tr 4063 df-iord 4326 df-suc 4331 |
This theorem is referenced by: suceloni 4460 ordsucg 4461 onsucsssucr 4468 ordtriexmidlem 4478 2ordpr 4483 ordsuc 4522 nnsucsssuc 6439 |
Copyright terms: Public domain | W3C validator |