Theorem inf2 8520

Description: Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 8536 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)

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 C_ 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 8519	. 2 |- E. x ( x =/= (/) /\ A. y ( y e. x -> E. z ( y e. z /\ z e. x ) ) )
3		dfss2 3591	. . . . 5 |- ( x C_ U. x <-> A. y ( y e. x -> y e. U. x ) )
4		eluni 4439	. . . . . . 7 |- ( y e. U. x <-> E. z ( y e. z /\ z e. x ) )
5	4	imbi2i 326	. . . . . 6 |- ( ( y e. x -> y e. U. x ) <-> ( y e. x -> E. z ( y e. z /\ z e. x ) ) )
6	5	albii 1747	. . . . 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	bitri 264	. . . 4 |- ( x C_ U. x <-> A. y ( y e. x -> E. z ( y e. z /\ z e. x ) ) )
8	7	anbi2i 730	. . 3 |- ( ( x =/= (/) /\ x C_ U. x ) <-> ( x =/= (/) /\ A. y ( y e. x -> E. z ( y e. z /\ z e. x ) ) ) )
9	8	exbii 1774	. 2 |- ( E. x ( x =/= (/) /\ x C_ U. x ) <-> E. x ( x =/= (/) /\ A. y ( y e. x -> E. z ( y e. z /\ z e. x ) ) ) )
10	2, 9	mpbir 221	1 |- E. x ( x =/= (/) /\ x C_ U. x )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-uni 4437

This theorem is referenced by:  axinf2  8537