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Metamath Proof Explorer

Theorem infdif2

Description: Cardinality ordering for an infinite class difference. (Contributed by NM, 24-Mar-2007) (Revised by Mario Carneiro, 29-Apr-2015)

Ref Expression
Assertion infdif2 
|- ( ( A e. dom card /\ B e. dom card /\ _om ~<_ A ) -> ( ( A \ B ) ~<_ B <-> A ~<_ B ) )

Proof

Step Hyp Ref Expression
1 domnsym 
 |-  ( ( A \ B ) ~<_ B -> -. B ~< ( A \ B ) )
2 simp3 
 |-  ( ( A e. dom card /\ _om ~<_ A /\ B ~< A ) -> B ~< A )
3 infdif 
 |-  ( ( A e. dom card /\ _om ~<_ A /\ B ~< A ) -> ( A \ B ) ~~ A )
4 3 ensymd 
 |-  ( ( A e. dom card /\ _om ~<_ A /\ B ~< A ) -> A ~~ ( A \ B ) )
5 sdomentr 
 |-  ( ( B ~< A /\ A ~~ ( A \ B ) ) -> B ~< ( A \ B ) )
6 2 4 5 syl2anc 
 |-  ( ( A e. dom card /\ _om ~<_ A /\ B ~< A ) -> B ~< ( A \ B ) )
7 1 6 nsyl3 
 |-  ( ( A e. dom card /\ _om ~<_ A /\ B ~< A ) -> -. ( A \ B ) ~<_ B )
8 7 3expia 
 |-  ( ( A e. dom card /\ _om ~<_ A ) -> ( B ~< A -> -. ( A \ B ) ~<_ B ) )
9 8 3adant2 
 |-  ( ( A e. dom card /\ B e. dom card /\ _om ~<_ A ) -> ( B ~< A -> -. ( A \ B ) ~<_ B ) )
10 9 con2d 
 |-  ( ( A e. dom card /\ B e. dom card /\ _om ~<_ A ) -> ( ( A \ B ) ~<_ B -> -. B ~< A ) )
11 domtri2 
 |-  ( ( A e. dom card /\ B e. dom card ) -> ( A ~<_ B <-> -. B ~< A ) )
12 11 3adant3 
 |-  ( ( A e. dom card /\ B e. dom card /\ _om ~<_ A ) -> ( A ~<_ B <-> -. B ~< A ) )
13 10 12 sylibrd 
 |-  ( ( A e. dom card /\ B e. dom card /\ _om ~<_ A ) -> ( ( A \ B ) ~<_ B -> A ~<_ B ) )
14 simp1 
 |-  ( ( A e. dom card /\ B e. dom card /\ _om ~<_ A ) -> A e. dom card )
15 difss 
 |-  ( A \ B ) C_ A
16 ssdomg 
 |-  ( A e. dom card -> ( ( A \ B ) C_ A -> ( A \ B ) ~<_ A ) )
17 14 15 16 mpisyl 
 |-  ( ( A e. dom card /\ B e. dom card /\ _om ~<_ A ) -> ( A \ B ) ~<_ A )
18 domtr 
 |-  ( ( ( A \ B ) ~<_ A /\ A ~<_ B ) -> ( A \ B ) ~<_ B )
19 18 ex 
 |-  ( ( A \ B ) ~<_ A -> ( A ~<_ B -> ( A \ B ) ~<_ B ) )
20 17 19 syl 
 |-  ( ( A e. dom card /\ B e. dom card /\ _om ~<_ A ) -> ( A ~<_ B -> ( A \ B ) ~<_ B ) )
21 13 20 impbid 
 |-  ( ( A e. dom card /\ B e. dom card /\ _om ~<_ A ) -> ( ( A \ B ) ~<_ B <-> A ~<_ B ) )