Theorem trsspwALT3 16642

Description: Short predicate calculus proof of the left-to-right implication of dftr4 3416. A transitive class is a subset of its power class. This proof was constructed by applying Metamath's minimize command to the proof of trsspwALT2 16641, which is the virtual deduction proof trsspwALT 16640 without virtual deductions.

Assertion

Ref	Expression
trsspwALT3	|- (Tr A -> A C_ ~PA)

Proof of Theorem trsspwALT3

Step	Hyp	Ref	Expression
1		trss 3421	. . 3 |- (Tr A -> (x e. A -> x C_ A))
2		visset 2295	. . . 4 |- x e. _V
3	2	elpw 3037	. . 3 |- (x e. ~PA <-> x C_ A)
4	1, 3	syl6ibr 230	. 2 |- (Tr A -> (x e. A -> x e. ~PA))
5	4	ssrdv 2622	1 |- (Tr A -> A C_ ~PA)

Syntax hints:   -> wi 3   e. wcel 1300   C_ wss 2593  ~Pcpw 3032  Tr wtr 3411

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-in 2603  df-ss 2605  df-pw 3035  df-uni 3178  df-tr 3412

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