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Theorem suctr 6256
Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) (Proof shortened by JJ, 24-Sep-2021.)
Assertion
Ref Expression
suctr (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctr
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsuci 6239 . . . . . 6 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2 trel 5144 . . . . . . . 8 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
32expdimp 456 . . . . . . 7 ((Tr 𝐴𝑧𝑦) → (𝑦𝐴𝑧𝐴))
4 eleq2 2822 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
54biimpcd 252 . . . . . . . 8 (𝑧𝑦 → (𝑦 = 𝐴𝑧𝐴))
65adantl 485 . . . . . . 7 ((Tr 𝐴𝑧𝑦) → (𝑦 = 𝐴𝑧𝐴))
73, 6jaod 858 . . . . . 6 ((Tr 𝐴𝑧𝑦) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧𝐴))
81, 7syl5 34 . . . . 5 ((Tr 𝐴𝑧𝑦) → (𝑦 ∈ suc 𝐴𝑧𝐴))
98expimpd 457 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝐴))
10 elelsuc 6245 . . . 4 (𝑧𝐴𝑧 ∈ suc 𝐴)
119, 10syl6 35 . . 3 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1211alrimivv 1935 . 2 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
13 dftr2 5139 . 2 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1412, 13sylibr 237 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 846  wal 1540   = wceq 1542  wcel 2114  Tr wtr 5137  suc csuc 6175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-v 3401  df-un 3849  df-in 3851  df-ss 3861  df-sn 4518  df-uni 4798  df-tr 5138  df-suc 6179
This theorem is referenced by:  dfon2lem3  33338  dfon2lem7  33342  dford3lem2  40444
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