Theorem inf2 4588

Description: Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using axinf 4603 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 4587	. 2 |- E.x(x =/= (/) /\ A.y(y e. x -> E.z(y e. z /\ z e. x)))
3		dfss2 2054	. . . . 5 |- (x (_ U.x <-> A.y(y e. x -> y e. U.x))
4		eluni 2501	. . . . . . 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 997	. . . . 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 1049	. 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 952   e. wcel 956  E.wex 978   =/= wne 1582   (_ wss 2043  (/)c0 2276  U.cuni 2498

This theorem is referenced by:  axinf2 4604  grothinf 8720

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-in 2047  df-ss 2049  df-nul 2277  df-uni 2499

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