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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 5791 | . . . . . 6 ⊢ (𝑦 ∈ suc 𝐴 → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)) | |
2 | trel 4759 | . . . . . . . 8 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) | |
3 | 2 | expdimp 453 | . . . . . . 7 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)) |
4 | eleq2 2690 | . . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴)) | |
5 | 4 | biimpcd 239 | . . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴)) |
6 | 5 | adantl 482 | . . . . . . 7 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴)) |
7 | 3, 6 | jaod 395 | . . . . . 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 629 | . . . 4 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝐴)) |
10 | elelsuc 5797 | . . . 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 1856 | . 2 ⊢ (Tr 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) |
13 | dftr2 4754 | . 2 ⊢ (Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) | |
14 | 12, 13 | sylibr 224 | 1 ⊢ (Tr 𝐴 → Tr suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 ∀wal 1481 = wceq 1483 ∈ wcel 1990 Tr wtr 4752 suc csuc 5725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 df-in 3581 df-ss 3588 df-sn 4178 df-uni 4437 df-tr 4753 df-suc 5729 |
This theorem is referenced by: dfon2lem3 31690 dfon2lem7 31694 dford3lem2 37594 |
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