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Theorem dftr5 4090
Description: An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
Assertion
Ref Expression
dftr5 (Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dftr5
StepHypRef Expression
1 dftr2 4089 . 2 (Tr 𝐴 ↔ ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2 alcom 1471 . . 3 (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
3 impexp 261 . . . . . . . 8 (((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ (𝑦𝑥 → (𝑥𝐴𝑦𝐴)))
43albii 1463 . . . . . . 7 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑦(𝑦𝑥 → (𝑥𝐴𝑦𝐴)))
5 df-ral 2453 . . . . . . 7 (∀𝑦𝑥 (𝑥𝐴𝑦𝐴) ↔ ∀𝑦(𝑦𝑥 → (𝑥𝐴𝑦𝐴)))
64, 5bitr4i 186 . . . . . 6 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑦𝑥 (𝑥𝐴𝑦𝐴))
7 r19.21v 2547 . . . . . 6 (∀𝑦𝑥 (𝑥𝐴𝑦𝐴) ↔ (𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
86, 7bitri 183 . . . . 5 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ (𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
98albii 1463 . . . 4 (∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
10 df-ral 2453 . . . 4 (∀𝑥𝐴𝑦𝑥 𝑦𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
119, 10bitr4i 186 . . 3 (∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
122, 11bitri 183 . 2 (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
131, 12bitri 183 1 (Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1346  wcel 2141  wral 2448  Tr wtr 4087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088
This theorem is referenced by:  dftr3  4091  exmidonfinlem  7170
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