Theorem cncnplem1 7774

Description: Lemma for cncnp2 7779.

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 2120	. . . 4 |- {x | (x e. A /\ (ph /\ x (_ B))} (_ {x | x (_ B}
3		df-rab 1652	. . . 4 |- {x e. A | (ph /\ x (_ B)} = {x | (x e. A /\ (ph /\ x (_ B))}
4		df-pw 2402	. . . 4 |- P~B = {x | x (_ B}
5	2, 3, 4	3sstr4 2100	. . 3 |- {x e. A | (ph /\ x (_ B)} (_ P~B
6		uniss 2521	. . 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 2756	. 2 |- U.P~B = B
9	7, 8	sseqtr 2093	1 |- U.{x e. A | (ph /\ x (_ B)} (_ B

Syntax hints:   /\ wa 223   e. wcel 958  {cab 1463  {crab 1648   (_ wss 2047  P~cpw 2401  U.cuni 2503

This theorem is referenced by:  cncnplem4 7777

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-uni 2504

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