Theorem trsspwALT 16640

Description: Virtual deduction proof of the left-to-right implication of dftr4 3416. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 3416 without accumulating results.

Assertion

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

Proof of Theorem trsspwALT

Step	Hyp	Ref	Expression
1		dfss2 2610	. . 3 |- (A C_ ~PA <-> A.x(x e. A -> x e. ~PA))
2		idn1 16484	. . . . . . 7 |- . Tr A   ⊢   Tr A .
3		idn2 16509	. . . . . . 7 |- . Tr A, x e. A   ⊢   x e. A .
4		trss 3421	. . . . . . 7 |- (Tr A -> (x e. A -> x C_ A))
5	2, 3, 4	e12 16593	. . . . . 6 |- . Tr A, x e. A   ⊢   x C_ A .
6		visset 2295	. . . . . . 7 |- x e. _V
7	6	elpw 3037	. . . . . 6 |- (x e. ~PA <-> x C_ A)
8	5, 7	e2bir 16523	. . . . 5 |- . Tr A, x e. A   ⊢   x e. ~PA .
9	8	in2 16506	. . . 4 |- . Tr A   ⊢   (x e. A -> x e. ~PA) .
10	9	gen11 16511	. . 3 |- . Tr A   ⊢   A.x(x e. A -> x e. ~PA) .
11		bi2 166	. . 3 |- ((A C_ ~PA <-> A.x(x e. A -> x e. ~PA)) -> (A.x(x e. A -> x e. ~PA) -> A C_ ~PA))
12	1, 10, 11	e01 16581	. 2 |- . Tr A   ⊢   A C_ ~PA .
13	12	in1 16481	1 |- (Tr A -> A C_ ~PA)

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   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  df-vd1 16480  df-vd2 16489

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