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Mirrors > Home > ILE Home > Th. List > onsucelsucr | GIF version |
Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4374. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6292. (Contributed by Jim Kingdon, 17-Jul-2019.) |
Ref | Expression |
---|---|
onsucelsucr | ⊢ (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2644 | . . . 4 ⊢ (suc 𝐴 ∈ suc 𝐵 → suc 𝐴 ∈ V) | |
2 | sucexb 4342 | . . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
3 | 1, 2 | sylibr 133 | . . 3 ⊢ (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ V) |
4 | onelss 4238 | . . . . . . 7 ⊢ (𝐵 ∈ On → (suc 𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) | |
5 | eqimss 3093 | . . . . . . . 8 ⊢ (suc 𝐴 = 𝐵 → suc 𝐴 ⊆ 𝐵) | |
6 | 5 | a1i 9 | . . . . . . 7 ⊢ (𝐵 ∈ On → (suc 𝐴 = 𝐵 → suc 𝐴 ⊆ 𝐵)) |
7 | 4, 6 | jaod 675 | . . . . . 6 ⊢ (𝐵 ∈ On → ((suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴 ⊆ 𝐵)) |
8 | 7 | adantl 272 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ On) → ((suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴 ⊆ 𝐵)) |
9 | elsucg 4255 | . . . . . . 7 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵))) | |
10 | 2, 9 | sylbi 120 | . . . . . 6 ⊢ (𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵))) |
11 | 10 | adantr 271 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵))) |
12 | eloni 4226 | . . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
13 | ordelsuc 4350 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)) | |
14 | 12, 13 | sylan2 281 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)) |
15 | 8, 11, 14 | 3imtr4d 202 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ 𝐵)) |
16 | 15 | impancom 257 | . . 3 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 ∈ suc 𝐵) → (𝐵 ∈ On → 𝐴 ∈ 𝐵)) |
17 | 3, 16 | mpancom 414 | . 2 ⊢ (suc 𝐴 ∈ suc 𝐵 → (𝐵 ∈ On → 𝐴 ∈ 𝐵)) |
18 | 17 | com12 30 | 1 ⊢ (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 667 = wceq 1296 ∈ wcel 1445 Vcvv 2633 ⊆ wss 3013 Ord word 4213 Oncon0 4214 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-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 |
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-pw 3451 df-sn 3472 df-pr 3473 df-uni 3676 df-tr 3959 df-iord 4217 df-on 4219 df-suc 4222 |
This theorem is referenced by: nnsucelsuc 6292 |
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