Theorem cncnplem1 7713

Description: Lemma for cncnp2 7718.

Assertion

Ref	Expression
cncnplem1	|- U.{x e. A | (ph /\ x (_ B)} (_ B

Distinct variable group:   x,B

Proof of Theorem cncnplem1

Step	Hyp	Ref	Expression
1		simprr 415	. . . . 5 |- ((x e. A /\ (ph /\ x (_ B)) -> x (_ B)
2	1	ss2abi 2110	. . . 4 |- {x | (x e. A /\ (ph /\ x (_ B))} (_ {x | x (_ B}
3		df-rab 1644	. . . 4 |- {x e. A | (ph /\ x (_ B)} = {x | (x e. A /\ (ph /\ x (_ B))}
4		df-pw 2392	. . . 4 |- P~B = {x | x (_ B}
5	2, 3, 4	3sstr4 2090	. . 3 |- {x e. A | (ph /\ x (_ B)} (_ P~B
6		uniss 2511	. . 3 |- ({x e. A | (ph /\ x (_ B)} (_ P~B -> U.{x e. A | (ph /\ x (_ B)} (_ U.P~B)
7	5, 6	ax-mp 7	. 2 |- U.{x e. A | (ph /\ x (_ B)} (_ U.P~B
8		unipw 2746	. 2 |- U.P~B = B
9	7, 8	sseqtr 2083	1 |- U.{x e. A | (ph /\ x (_ B)} (_ B

Syntax hints:   /\ wa 223   e. wcel 955  {cab 1456  {crab 1640   (_ wss 2037  P~cpw 2391  U.cuni 2493

This theorem is referenced by:  cncnplem4 7716

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-uni 2494

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