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Metamath Proof Explorer

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 )