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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 4308 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
2 | suctr 4351 | . . 3 ⊢ (Tr 𝐴 → Tr suc 𝐴) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (Ord 𝐴 → Tr suc 𝐴) |
4 | df-suc 4301 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | 4 | eleq2i 2207 | . . . . 5 ⊢ (𝑥 ∈ suc 𝐴 ↔ 𝑥 ∈ (𝐴 ∪ {𝐴})) |
6 | elun 3222 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴})) | |
7 | velsn 3549 | . . . . . 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 4297 | . . . . . . . 8 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) | |
11 | 10 | simprbi 273 | . . . . . . 7 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 Tr 𝑥) |
12 | df-ral 2422 | . . . . . . 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 4040 | . . . . . 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 2507 | . 2 ⊢ (Ord 𝐴 → ∀𝑥 ∈ suc 𝐴Tr 𝑥) |
20 | dford3 4297 | . 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 1330 = wceq 1332 ∈ wcel 1481 ∀wral 2417 ∪ cun 3074 {csn 3532 Tr wtr 4034 Ord word 4292 suc csuc 4295 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-uni 3745 df-tr 4035 df-iord 4296 df-suc 4301 |
This theorem is referenced by: suceloni 4425 ordsucg 4426 onsucsssucr 4433 ordtriexmidlem 4443 2ordpr 4447 ordsuc 4486 nnsucsssuc 6396 |
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