Metamath Proof Explorer


Theorem atsseq

Description: Two atoms in a subset relationship are equal. (Contributed by NM, 26-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion atsseq
|- ( ( A e. HAtoms /\ B e. HAtoms ) -> ( A C_ B <-> A = B ) )

Proof

Step Hyp Ref Expression
1 atne0
 |-  ( A e. HAtoms -> A =/= 0H )
2 1 ad2antrr
 |-  ( ( ( A e. HAtoms /\ B e. HAtoms ) /\ A C_ B ) -> A =/= 0H )
3 atelch
 |-  ( A e. HAtoms -> A e. CH )
4 atss
 |-  ( ( A e. CH /\ B e. HAtoms ) -> ( A C_ B -> ( A = B \/ A = 0H ) ) )
5 3 4 sylan
 |-  ( ( A e. HAtoms /\ B e. HAtoms ) -> ( A C_ B -> ( A = B \/ A = 0H ) ) )
6 5 imp
 |-  ( ( ( A e. HAtoms /\ B e. HAtoms ) /\ A C_ B ) -> ( A = B \/ A = 0H ) )
7 6 ord
 |-  ( ( ( A e. HAtoms /\ B e. HAtoms ) /\ A C_ B ) -> ( -. A = B -> A = 0H ) )
8 7 necon1ad
 |-  ( ( ( A e. HAtoms /\ B e. HAtoms ) /\ A C_ B ) -> ( A =/= 0H -> A = B ) )
9 2 8 mpd
 |-  ( ( ( A e. HAtoms /\ B e. HAtoms ) /\ A C_ B ) -> A = B )
10 9 ex
 |-  ( ( A e. HAtoms /\ B e. HAtoms ) -> ( A C_ B -> A = B ) )
11 eqimss
 |-  ( A = B -> A C_ B )
12 10 11 impbid1
 |-  ( ( A e. HAtoms /\ B e. HAtoms ) -> ( A C_ B <-> A = B ) )