Theorem pwtr 4263

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 4261	. . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴
2	1	sseq1i 3219	. 2 ⊢ (∪ 𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)
3		df-tr 4143	. 2 ⊢ (Tr 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝒫 𝐴)
4		dftr4 4147	. 2 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)
5	2, 3, 4	3bitr4ri 213	1 ⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴)

Syntax hints:   ↔ wb 105   ⊆ wss 3166  𝒫 cpw 3616  ∪ cuni 3850  Tr wtr 4142

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-uni 3851  df-tr 4143

This theorem is referenced by: (None)