Theorem abssexgOLD 3491

Description: Existence of a class of subsets.

Assertion

Ref	Expression
abssexgOLD	|- (A e. B -> {x | (x C_ A /\ ph)} e. _V)

Distinct variable group:   x,A

Proof of Theorem abssexgOLD

Step	Hyp	Ref	Expression
1		pwexg 3489	. . 3 |- (A e. B -> ~PA e. _V)
2		rabexg 3460	. . 3 |- (~PA e. _V -> {x e. ~PA | ph} e. _V)
3	1, 2	syl 12	. 2 |- (A e. B -> {x e. ~PA | ph} e. _V)
4		df-rab 2112	. . 3 |- {x e. ~PA | ph} = {x | (x e. ~PA /\ ph)}
5		visset 2295	. . . . . 6 |- x e. _V
6	5	elpw 3037	. . . . 5 |- (x e. ~PA <-> x C_ A)
7	6	anbi1i 539	. . . 4 |- ((x e. ~PA /\ ph) <-> (x C_ A /\ ph))
8	7	abbii 2006	. . 3 |- {x | (x e. ~PA /\ ph)} = {x | (x C_ A /\ ph)}
9	4, 8	eqtr2i 1909	. 2 |- {x | (x C_ A /\ ph)} = {x e. ~PA | ph}
10	3, 9	syl5eqel 1975	1 |- (A e. B -> {x | (x C_ A /\ ph)} e. _V)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  {cab 1871  {crab 2108  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032

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-14 1312  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  ax-sep 3438  ax-pow 3481

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-rab 2112  df-v 2294  df-in 2603  df-ss 2605  df-pw 3035

Copyright terms: Public domain