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Metamath Proof Explorer

Theorem ipolt

Description: Strict order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015)

Ref Expression
Hypotheses ipolt.i 
|- I = ( toInc ` F )
ipolt.l 
|- .< = ( lt ` I )
Assertion ipolt 
|- ( ( F e. V /\ X e. F /\ Y e. F ) -> ( X .< Y <-> X C. Y ) )

Proof

Step Hyp Ref Expression
1 ipolt.i 
 |-  I = ( toInc ` F )
2 ipolt.l 
 |-  .< = ( lt ` I )
3 eqid 
 |-  ( le ` I ) = ( le ` I )
4 1 3 ipole 
 |-  ( ( F e. V /\ X e. F /\ Y e. F ) -> ( X ( le ` I ) Y <-> X C_ Y ) )
5 4 anbi1d 
 |-  ( ( F e. V /\ X e. F /\ Y e. F ) -> ( ( X ( le ` I ) Y /\ X =/= Y ) <-> ( X C_ Y /\ X =/= Y ) ) )
6 1 fvexi 
 |-  I e. _V
7 3 2 pltval 
 |-  ( ( I e. _V /\ X e. F /\ Y e. F ) -> ( X .< Y <-> ( X ( le ` I ) Y /\ X =/= Y ) ) )
8 6 7 mp3an1 
 |-  ( ( X e. F /\ Y e. F ) -> ( X .< Y <-> ( X ( le ` I ) Y /\ X =/= Y ) ) )
9 8 3adant1 
 |-  ( ( F e. V /\ X e. F /\ Y e. F ) -> ( X .< Y <-> ( X ( le ` I ) Y /\ X =/= Y ) ) )
10 df-pss 
 |-  ( X C. Y <-> ( X C_ Y /\ X =/= Y ) )
11 10 a1i 
 |-  ( ( F e. V /\ X e. F /\ Y e. F ) -> ( X C. Y <-> ( X C_ Y /\ X =/= Y ) ) )
12 5 9 11 3bitr4d 
 |-  ( ( F e. V /\ X e. F /\ Y e. F ) -> ( X .< Y <-> X C. Y ) )