Theorem inf2 4617

Description: Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using axinf 4632 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 4616	. 2 |- E.x(x =/= (/) /\ A.y(y e. x -> E.z(y e. z /\ z e. x)))
3		dfss2 2061	. . . . 5 |- (x (_ U.x <-> A.y(y e. x -> y e. U.x))
4		eluni 2510	. . . . . . 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 1001	. . . . 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 482	. . 3 |- ((x =/= (/) /\ x (_ U.x) <-> (x =/= (/) /\ A.y(y e. x -> E.z(y e. z /\ z e. x))))
9	8	exbii 1053	. 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 956   e. wcel 960  E.wex 982   =/= wne 1588   (_ wss 2050  (/)c0 2283  U.cuni 2507

This theorem is referenced by:  axinf2 4633  grothinf 8776

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284  df-uni 2508

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