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Mirrors > Home > MPE Home > Th. List > suctr | Structured version Visualization version GIF version |
Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) (Proof shortened by JJ, 24-Sep-2021.) |
Ref | Expression |
---|---|
suctr | ⊢ (Tr 𝐴 → Tr suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsuci 6225 | . . . . . 6 ⊢ (𝑦 ∈ suc 𝐴 → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)) | |
2 | trel 5143 | . . . . . . . 8 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) | |
3 | 2 | expdimp 456 | . . . . . . 7 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)) |
4 | eleq2 2878 | . . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴)) | |
5 | 4 | biimpcd 252 | . . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴)) |
6 | 5 | adantl 485 | . . . . . . 7 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴)) |
7 | 3, 6 | jaod 856 | . . . . . 6 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → 𝑧 ∈ 𝐴)) |
8 | 1, 7 | syl5 34 | . . . . 5 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → (𝑦 ∈ suc 𝐴 → 𝑧 ∈ 𝐴)) |
9 | 8 | expimpd 457 | . . . 4 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝐴)) |
10 | elelsuc 6231 | . . . 4 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) | |
11 | 9, 10 | syl6 35 | . . 3 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) |
12 | 11 | alrimivv 1929 | . 2 ⊢ (Tr 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) |
13 | dftr2 5138 | . 2 ⊢ (Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) | |
14 | 12, 13 | sylibr 237 | 1 ⊢ (Tr 𝐴 → Tr suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 ∀wal 1536 = wceq 1538 ∈ wcel 2111 Tr wtr 5136 suc csuc 6161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-uni 4801 df-tr 5137 df-suc 6165 |
This theorem is referenced by: dfon2lem3 33143 dfon2lem7 33147 dford3lem2 39968 |
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