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Metamath Proof Explorer

Theorem inf3lem4

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 for detailed description. (Contributed by NM, 29-Oct-1996)

Ref Expression
Hypotheses inf3lem.1 
|- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } )
inf3lem.2 
|- F = ( rec ( G , (/) ) |` _om )
inf3lem.3 
|- A e. _V
inf3lem.4 
|- B e. _V
Assertion inf3lem4 
|- ( ( x =/= (/) /\ x C_ U. x ) -> ( A e. _om -> ( F ` A ) C. ( F ` suc A ) ) )

Proof

Step Hyp Ref Expression
1 inf3lem.1 
 |-  G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } )
2 inf3lem.2 
 |-  F = ( rec ( G , (/) ) |` _om )
3 inf3lem.3 
 |-  A e. _V
4 inf3lem.4 
 |-  B e. _V
5 1 2 3 4 inf3lem1 
 |-  ( A e. _om -> ( F ` A ) C_ ( F ` suc A ) )
6 5 a1i 
 |-  ( ( x =/= (/) /\ x C_ U. x ) -> ( A e. _om -> ( F ` A ) C_ ( F ` suc A ) ) )
7 1 2 3 4 inf3lem3 
 |-  ( ( x =/= (/) /\ x C_ U. x ) -> ( A e. _om -> ( F ` A ) =/= ( F ` suc A ) ) )
8 6 7 jcad 
 |-  ( ( x =/= (/) /\ x C_ U. x ) -> ( A e. _om -> ( ( F ` A ) C_ ( F ` suc A ) /\ ( F ` A ) =/= ( F ` suc A ) ) ) )
9 df-pss 
 |-  ( ( F ` A ) C. ( F ` suc A ) <-> ( ( F ` A ) C_ ( F ` suc A ) /\ ( F ` A ) =/= ( F ` suc A ) ) )
10 8 9 imbitrrdi 
 |-  ( ( x =/= (/) /\ x C_ U. x ) -> ( A e. _om -> ( F ` A ) C. ( F ` suc A ) ) )