Metamath Proof Explorer

Theorem ramtcl

Description: The Ramsey number has the Ramsey number property if any number does. (Contributed by Mario Carneiro, 20-Apr-2015) (Revised by AV, 14-Sep-2020)

Ref Expression
Hypotheses ramval.c 
|- C = ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } )
ramval.t 
|- T = { n e. NN0 | A. s ( n <_ ( # ` s ) -> A. f e. ( R ^m ( s C M ) ) E. c e. R E. x e. ~P s ( ( F ` c ) <_ ( # ` x ) /\ ( x C M ) C_ ( `' f " { c } ) ) ) }
Assertion ramtcl 
|- ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) -> ( ( M Ramsey F ) e. T <-> T =/= (/) ) )

Proof

Step Hyp Ref Expression
1 ramval.c 
 |-  C = ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } )
2 ramval.t 
 |-  T = { n e. NN0 | A. s ( n <_ ( # ` s ) -> A. f e. ( R ^m ( s C M ) ) E. c e. R E. x e. ~P s ( ( F ` c ) <_ ( # ` x ) /\ ( x C M ) C_ ( `' f " { c } ) ) ) }
3 ne0i 
 |-  ( ( M Ramsey F ) e. T -> T =/= (/) )
4 1 2 ramcl2lem 
 |-  ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) -> ( M Ramsey F ) = if ( T = (/) , +oo , inf ( T , RR , < ) ) )
5 ifnefalse 
 |-  ( T =/= (/) -> if ( T = (/) , +oo , inf ( T , RR , < ) ) = inf ( T , RR , < ) )
6 4 5 sylan9eq 
 |-  ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> ( M Ramsey F ) = inf ( T , RR , < ) )
7 2 ssrab3 
 |-  T C_ NN0
8 nn0uz 
 |-  NN0 = ( ZZ>= ` 0 )
9 7 8 sseqtri 
 |-  T C_ ( ZZ>= ` 0 )
10 9 a1i 
 |-  ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) -> T C_ ( ZZ>= ` 0 ) )
11 infssuzcl 
 |-  ( ( T C_ ( ZZ>= ` 0 ) /\ T =/= (/) ) -> inf ( T , RR , < ) e. T )
12 10 11 sylan 
 |-  ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> inf ( T , RR , < ) e. T )
13 6 12 eqeltrd 
 |-  ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> ( M Ramsey F ) e. T )
14 13 ex 
 |-  ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) -> ( T =/= (/) -> ( M Ramsey F ) e. T ) )
15 3 14 impbid2 
 |-  ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) -> ( ( M Ramsey F ) e. T <-> T =/= (/) ) )