Theorem pwtr 5348

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 5346	. . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴
2	1	sseq1i 3998	. 2 ⊢ (∪ 𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)
3		df-tr 5176	. 2 ⊢ (Tr 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝒫 𝐴)
4		dftr4 5180	. 2 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)
5	2, 3, 4	3bitr4ri 306	1 ⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴)

Syntax hints:   ↔ wb 208   ⊆ wss 3939  𝒫 cpw 4542  ∪ cuni 4841  Tr wtr 5175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-pw 4544  df-sn 4571  df-pr 4573  df-uni 4842  df-tr 5176

This theorem is referenced by:  r1tr  9208  itunitc1  9845