Theorem dftr4 4215

Description: An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)

Assertion

Ref	Expression
dftr4	⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)

Proof of Theorem dftr4

Step	Hyp	Ref	Expression
1		df-tr 4211	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
2		sspwuni 4078	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
3	1, 2	bitr4i 187	1 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)

Syntax hints:   ↔ wb 105   ⊆ wss 3213  𝒫 cpw 3671  ∪ cuni 3916  Tr wtr 4210

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-in 3219  df-ss 3226  df-pw 3673  df-uni 3917  df-tr 4211

This theorem is referenced by:  tr0  4221  pwtr  4337  pw1on  7538