Theorem inf2 3459

Description: Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using axinf 1084 as a hypothesis).

Hypothesis

Ref	Expression
inf1.1	|- E.x(y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x)))

Assertion

Ref	Expression
inf2	|- E.x(-. x = (/) /\ x (_ U.x)

Distinct variable group(s):   x,y,z

Proof of Theorem inf2

Step	Hyp	Ref	Expression
1		inf1.1	. . 3 |- E.x(y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x)))
2	1	inf1 3458	. 2 |- E.x(-. x = (/) /\ A.y(y e. x -> E.z(y e. z /\ z e. x)))
3		dfss2 1497	. . . . 5 |- (x (_ U.x <-> A.y(y e. x -> y e. U.x))
4		eluni 1922	. . . . . . 7 |- (y e. U.x <-> E.z(y e. z /\ z e. x))
5	4	imbi2i 160	. . . . . 6 |- ((y e. x -> y e. U.x) <-> (y e. x -> E.z(y e. z /\ z e. x)))
6	5	bial 695	. . . . 5 |- (A.y(y e. x -> y e. U.x) <-> A.y(y e. x -> E.z(y e. z /\ z e. x)))
7	3, 6	bitr 151	. . . 4 |- (x (_ U.x <-> A.y(y e. x -> E.z(y e. z /\ z e. x)))
8	7	anbi2i 367	. . 3 |- ((-. x = (/) /\ x (_ U.x) <-> (-. x = (/) /\ A.y(y e. x -> E.z(y e. z /\ z e. x))))
9	8	biex 733	. 2 |- (E.x(-. x = (/) /\ x (_ U.x) <-> E.x(-. x = (/) /\ A.y(y e. x -> E.z(y e. z /\ z e. x))))
10	2, 9	mpbir 165	1 |- E.x(-. x = (/) /\ x (_ U.x)

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   e. wel 803   = wceq 1091   e. wcel 1092   (_ wss 1487  (/)c0 1707  U.cuni 1919

This theorem is referenced by:  inf4 3473

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-uni 1920

metamath.org