Theorem cncnplem2 7775

Description: Lemma for cncnp2 7779.

Assertion

Ref	Expression
cncnplem2	|- (A.x(x e. A -> x e. B) -> A (_ U_x e. A B)

Distinct variable group:   x,A

Proof of Theorem cncnplem2

Step	Hyp	Ref	Expression
1		ax-17 971	. . . . . 6 |- (y e. A -> A.x y e. A)
2		hba1 1003	. . . . . . 7 |- (A.x(x e. A -> x e. B) -> A.xA.x(x e. A -> x e. B))
3		hbre1 1689	. . . . . . 7 |- (E.x e. A y e. B -> A.xE.x e. A y e. B)
4	2, 3	hbim 1007	. . . . . 6 |- ((A.x(x e. A -> x e. B) -> E.x e. A y e. B) -> A.x(A.x(x e. A -> x e. B) -> E.x e. A y e. B))
5	1, 4	hbim 1007	. . . . 5 |- ((y e. A -> (A.x(x e. A -> x e. B) -> E.x e. A y e. B)) -> A.x(y e. A -> (A.x(x e. A -> x e. B) -> E.x e. A y e. B)))
6		visset 1813	. . . . 5 |- y e. V
7		ra4e 1695	. . . . . . 7 |- ((x e. A /\ y e. B) -> E.x e. A y e. B)
8		eleq1 1534	. . . . . . . . 9 |- (x = y -> (x e. A <-> y e. A))
9	8	biimpar 417	. . . . . . . 8 |- ((x = y /\ y e. A) -> x e. A)
10	9	adantr 389	. . . . . . 7 |- (((x = y /\ y e. A) /\ A.x(x e. A -> x e. B)) -> x e. A)
11		ax-4 973	. . . . . . . . . . 11 |- (A.x(x e. A -> x e. B) -> (x e. A -> x e. B))
12	11	com12 11	. . . . . . . . . 10 |- (x e. A -> (A.x(x e. A -> x e. B) -> x e. B))
13	8, 12	syl6bir 215	. . . . . . . . 9 |- (x = y -> (y e. A -> (A.x(x e. A -> x e. B) -> x e. B)))
14		eleq1 1534	. . . . . . . . . 10 |- (x = y -> (x e. B <-> y e. B))
15	14	biimpd 153	. . . . . . . . 9 |- (x = y -> (x e. B -> y e. B))
16	13, 15	syl6d 56	. . . . . . . 8 |- (x = y -> (y e. A -> (A.x(x e. A -> x e. B) -> y e. B)))
17	16	imp31 362	. . . . . . 7 |- (((x = y /\ y e. A) /\ A.x(x e. A -> x e. B)) -> y e. B)
18	7, 10, 17	sylanc 471	. . . . . 6 |- (((x = y /\ y e. A) /\ A.x(x e. A -> x e. B)) -> E.x e. A y e. B)
19	18	exp31 376	. . . . 5 |- (x = y -> (y e. A -> (A.x(x e. A -> x e. B) -> E.x e. A y e. B)))
20	5, 6, 19	vtoclef 1857	. . . 4 |- (y e. A -> (A.x(x e. A -> x e. B) -> E.x e. A y e. B))
21		eliun 2570	. . . 4 |- (y e. U_x e. A B <-> E.x e. A y e. B)
22	20, 21	syl6ibr 213	. . 3 |- (y e. A -> (A.x(x e. A -> x e. B) -> y e. U_x e. A B))
23	22	com12 11	. 2 |- (A.x(x e. A -> x e. B) -> (y e. A -> y e. U_x e. A B))
24	23	ssrdv 2070	1 |- (A.x(x e. A -> x e. B) -> A (_ U_x e. A B)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wrex 1646   (_ wss 2047  U_ciun 2566

This theorem is referenced by:  cncnplem3 7776

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-12 968  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

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1650  df-v 1812  df-in 2051  df-ss 2053  df-iun 2568

Copyright terms: Public domain