Theorem infeq5

Description: The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex .) The left-hand side provides us with a very short way to express the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity. (Contributed by NM, 23-Mar-2004) (Revised by Mario Carneiro, 16-Nov-2014)

	Ref	Expression
Assertion	infeq5	|- ( E. x x C. U. x <-> _om e. _V )

Proof

Step	Hyp	Ref	Expression
1		df-pss	|- ( x C. U. x <-> ( x C_ U. x /\ x =/= U. x ) )
2		unieq	|- ( x = (/) -> U. x = U. (/) )
3		uni0	|- U. (/) = (/)
4	2 3	eqtr2di	|- ( x = (/) -> (/) = U. x )
5		eqtr	|- ( ( x = (/) /\ (/) = U. x ) -> x = U. x )
6	4 5	mpdan	|- ( x = (/) -> x = U. x )
7	6	necon3i	|- ( x =/= U. x -> x =/= (/) )
8	7	anim1i	|- ( ( x =/= U. x /\ x C_ U. x ) -> ( x =/= (/) /\ x C_ U. x ) )
9	8	ancoms	|- ( ( x C_ U. x /\ x =/= U. x ) -> ( x =/= (/) /\ x C_ U. x ) )
10	1 9	sylbi	|- ( x C. U. x -> ( x =/= (/) /\ x C_ U. x ) )
11	10	eximi	|- ( E. x x C. U. x -> E. x ( x =/= (/) /\ x C_ U. x ) )
12		eqid	|- ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } )
13		eqid	|- ( rec ( ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) , (/) ) |` _om ) = ( rec ( ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) , (/) ) |` _om )
14		vex	|- x e. _V
15	12 13 14 14	inf3lem7	|- ( ( x =/= (/) /\ x C_ U. x ) -> _om e. _V )
16	15	exlimiv	|- ( E. x ( x =/= (/) /\ x C_ U. x ) -> _om e. _V )
17	11 16	syl	|- ( E. x x C. U. x -> _om e. _V )
18		infeq5i	|- ( _om e. _V -> E. x x C. U. x )
19	17 18	impbii	|- ( E. x x C. U. x <-> _om e. _V )