Metamath Proof Explorer


Theorem ssmd2

Description: Ordering implies the modular pair property. Remark in MaedaMaeda p. 1. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion ssmd2 ACBCABB𝑀A

Proof

Step Hyp Ref Expression
1 inss2 xBAA
2 chub2 ACxCAxA
3 1 2 sstrid ACxCxBAxA
4 3 adantrl ACABxCxBAxA
5 simpl ABxCAB
6 sseqin2 ABBA=A
7 5 6 sylib ABxCBA=A
8 7 adantl ACABxCBA=A
9 8 oveq2d ACABxCxBA=xA
10 4 9 sseqtrrd ACABxCxBAxBA
11 10 a1d ACABxCxAxBAxBA
12 11 exp32 ACABxCxAxBAxBA
13 12 ralrimdv ACABxCxAxBAxBA
14 13 adantr ACBCABxCxAxBAxBA
15 mdbr2 BCACB𝑀AxCxAxBAxBA
16 15 ancoms ACBCB𝑀AxCxAxBAxBA
17 14 16 sylibrd ACBCABB𝑀A
18 17 3impia ACBCABB𝑀A