Theorem inf2 4608

Description: Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using axinf 4623 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283.

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:   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 4607	. 2 |- E.x(x =/= (/) /\ A.y(y e. x -> E.z(y e. z /\ z e. x)))
3		dfss2 2058	. . . . 5 |- (x (_ U.x <-> A.y(y e. x -> y e. U.x))
4		eluni 2506	. . . . . . 7 |- (y e. U.x <-> E.z(y e. z /\ z e. x))
5	4	imbi2i 185	. . . . . 6 |- ((y e. x -> y e. U.x) <-> (y e. x -> E.z(y e. z /\ z e. x)))
6	5	albii 999	. . . . 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 173	. . . 4 |- (x (_ U.x <-> A.y(y e. x -> E.z(y e. z /\ z e. x)))
8	7	anbi2i 480	. . 3 |- ((x =/= (/) /\ x (_ U.x) <-> (x =/= (/) /\ A.y(y e. x -> E.z(y e. z /\ z e. x))))
9	8	exbii 1051	. 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 190	1 |- E.x(x =/= (/) /\ x (_ U.x)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958  E.wex 980   =/= wne 1585   (_ wss 2047  (/)c0 2280  U.cuni 2503

This theorem is referenced by:  axinf2 4624  grothinf 8781

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-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281  df-uni 2504

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