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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 4541. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6506. (Contributed by Jim Kingdon, 17-Jul-2019.) |
Ref | Expression |
---|---|
onsucelsucr | ⊢ (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2760 | . . . 4 ⊢ (suc 𝐴 ∈ suc 𝐵 → suc 𝐴 ∈ V) | |
2 | sucexb 4508 | . . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
3 | 1, 2 | sylibr 134 | . . 3 ⊢ (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ V) |
4 | onelss 4399 | . . . . . . 7 ⊢ (𝐵 ∈ On → (suc 𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) | |
5 | eqimss 3221 | . . . . . . . 8 ⊢ (suc 𝐴 = 𝐵 → suc 𝐴 ⊆ 𝐵) | |
6 | 5 | a1i 9 | . . . . . . 7 ⊢ (𝐵 ∈ On → (suc 𝐴 = 𝐵 → suc 𝐴 ⊆ 𝐵)) |
7 | 4, 6 | jaod 718 | . . . . . 6 ⊢ (𝐵 ∈ On → ((suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴 ⊆ 𝐵)) |
8 | 7 | adantl 277 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ On) → ((suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴 ⊆ 𝐵)) |
9 | elsucg 4416 | . . . . . . 7 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵))) | |
10 | 2, 9 | sylbi 121 | . . . . . 6 ⊢ (𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵))) |
11 | 10 | adantr 276 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵))) |
12 | eloni 4387 | . . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
13 | ordelsuc 4516 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)) | |
14 | 12, 13 | sylan2 286 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)) |
15 | 8, 11, 14 | 3imtr4d 203 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ 𝐵)) |
16 | 15 | impancom 260 | . . 3 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 ∈ suc 𝐵) → (𝐵 ∈ On → 𝐴 ∈ 𝐵)) |
17 | 3, 16 | mpancom 422 | . 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 104 ↔ wb 105 ∨ wo 709 = wceq 1363 ∈ wcel 2158 Vcvv 2749 ⊆ wss 3141 Ord word 4374 Oncon0 4375 suc csuc 4377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-uni 3822 df-tr 4114 df-iord 4378 df-on 4380 df-suc 4383 |
This theorem is referenced by: nnsucelsuc 6506 |
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