Theorem pwtr 5414

Description: A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)

Assertion

Ref	Expression
pwtr	⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴)

Proof of Theorem pwtr

Step	Hyp	Ref	Expression
1		unipw 5412	. . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴
2	1	sseq1i 3975	. 2 ⊢ (∪ 𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)
3		df-tr 5228	. 2 ⊢ (Tr 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝒫 𝐴)
4		dftr4 5234	. 2 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)
5	2, 3, 4	3bitr4ri 303	1 ⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴)

Syntax hints:   ↔ wb 205   ⊆ wss 3913  𝒫 cpw 4565  ∪ cuni 4870  Tr wtr 5227

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-v 3448  df-un 3918  df-in 3920  df-ss 3930  df-pw 4567  df-sn 4592  df-pr 4594  df-uni 4871  df-tr 5228

This theorem is referenced by:  r1tr  9721  itunitc1  10365