Theorem abssexg 3490

Description: Existence of a class of subsets. (The proof was shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

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

Distinct variable group:   x,A

Proof of Theorem abssexg

Step	Hyp	Ref	Expression
1		pwexg 3489	. 2 |- (A e. B -> ~PA e. _V)
2		df-pw 3035	. . . 4 |- ~PA = {x | x C_ A}
3	2	eleq1i 1960	. . 3 |- (~PA e. _V <-> {x | x C_ A} e. _V)
4		simpl 346	. . . . 5 |- ((x C_ A /\ ph) -> x C_ A)
5	4	ss2abi 2679	. . . 4 |- {x | (x C_ A /\ ph)} C_ {x | x C_ A}
6		ssexg 3457	. . . 4 |- (({x | (x C_ A /\ ph)} C_ {x | x C_ A} /\ {x | x C_ A} e. _V) -> {x | (x C_ A /\ ph)} e. _V)
7	5, 6	mpan 759	. . 3 |- ({x | x C_ A} e. _V -> {x | (x C_ A /\ ph)} e. _V)
8	3, 7	sylbi 216	. 2 |- (~PA e. _V -> {x | (x C_ A /\ ph)} e. _V)
9	1, 8	syl 12	1 |- (A e. B -> {x | (x C_ A /\ ph)} e. _V)

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

This theorem is referenced by:  pmex 5386  tgval 8886  tgval3 8895  fctop 8920  cctop 8922  cldval 8942  neif 8991  neival 8993  opnfval 9134  caufval 9204  issubg 9425  subsp 10244  neibastop1 15518  fnemeet2 15529  supfil 15560  ufinffr 15578  ufilen 15579  flimfnfcls 15615

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

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