Theorem pwtr 4262

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 4260	. . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴
2	1	sseq1i 3218	. 2 ⊢ (∪ 𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)
3		df-tr 4142	. 2 ⊢ (Tr 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝒫 𝐴)
4		dftr4 4146	. 2 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)
5	2, 3, 4	3bitr4ri 213	1 ⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴)

Syntax hints:   ↔ wb 105   ⊆ wss 3165  𝒫 cpw 3615  ∪ cuni 3849  Tr wtr 4141

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-uni 3850  df-tr 4142

This theorem is referenced by: (None)