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Metamath Proof Explorer

Theorem hashdomi

Description: Non-strict order relation of the # function on the full cardinal poset. (Contributed by Stefan O'Rear, 12-Sep-2015)

Ref Expression
Assertion hashdomi 
|- ( A ~<_ B -> ( # ` A ) <_ ( # ` B ) )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( A ~<_ B /\ A e. Fin ) -> A ~<_ B )
2 simpr 
 |-  ( ( A ~<_ B /\ A e. Fin ) -> A e. Fin )
3 reldom 
 |-  Rel ~<_
4 3 brrelex2i 
 |-  ( A ~<_ B -> B e. _V )
5 4 adantr 
 |-  ( ( A ~<_ B /\ A e. Fin ) -> B e. _V )
6 hashdom 
 |-  ( ( A e. Fin /\ B e. _V ) -> ( ( # ` A ) <_ ( # ` B ) <-> A ~<_ B ) )
7 2 5 6 syl2anc 
 |-  ( ( A ~<_ B /\ A e. Fin ) -> ( ( # ` A ) <_ ( # ` B ) <-> A ~<_ B ) )
8 1 7 mpbird 
 |-  ( ( A ~<_ B /\ A e. Fin ) -> ( # ` A ) <_ ( # ` B ) )
9 pnfxr 
 |-  +oo e. RR*
10 pnfge 
 |-  ( +oo e. RR* -> +oo <_ +oo )
11 9 10 mp1i 
 |-  ( ( A ~<_ B /\ -. A e. Fin ) -> +oo <_ +oo )
12 3 brrelex1i 
 |-  ( A ~<_ B -> A e. _V )
13 hashinf 
 |-  ( ( A e. _V /\ -. A e. Fin ) -> ( # ` A ) = +oo )
14 12 13 sylan 
 |-  ( ( A ~<_ B /\ -. A e. Fin ) -> ( # ` A ) = +oo )
15 4 adantr 
 |-  ( ( A ~<_ B /\ -. A e. Fin ) -> B e. _V )
16 domfi 
 |-  ( ( B e. Fin /\ A ~<_ B ) -> A e. Fin )
17 16 stoic1b 
 |-  ( ( A ~<_ B /\ -. A e. Fin ) -> -. B e. Fin )
18 hashinf 
 |-  ( ( B e. _V /\ -. B e. Fin ) -> ( # ` B ) = +oo )
19 15 17 18 syl2anc 
 |-  ( ( A ~<_ B /\ -. A e. Fin ) -> ( # ` B ) = +oo )
20 11 14 19 3brtr4d 
 |-  ( ( A ~<_ B /\ -. A e. Fin ) -> ( # ` A ) <_ ( # ` B ) )
21 8 20 pm2.61dan 
 |-  ( A ~<_ B -> ( # ` A ) <_ ( # ` B ) )