Theorem pwtr 5437

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 5435	. . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴
2	1	sseq1i 3973	. 2 ⊢ (∪ 𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)
3		df-tr 5224	. 2 ⊢ (Tr 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝒫 𝐴)
4		dftr4 5229	. 2 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)
5	2, 3, 4	3bitr4ri 307	1 ⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴)

Syntax hints:   ↔ wb 209   ⊆ wss 3913  𝒫 cpw 4567  ∪ cuni 4877  Tr wtr 5223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-v 3464  df-un 3918  df-ss 3930  df-pw 4569  df-sn 4595  df-pr 4597  df-uni 4878  df-tr 5224

This theorem is referenced by:  r1tr  9751  itunitc1  10407  ttcpwss  36974