Metamath Proof Explorer


Theorem ltprord

Description: Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996) (New usage is discouraged.)

Ref Expression
Assertion ltprord A 𝑷 B 𝑷 A < 𝑷 B A B

Proof

Step Hyp Ref Expression
1 eleq1 x = A x 𝑷 A 𝑷
2 1 anbi1d x = A x 𝑷 y 𝑷 A 𝑷 y 𝑷
3 psseq1 x = A x y A y
4 2 3 anbi12d x = A x 𝑷 y 𝑷 x y A 𝑷 y 𝑷 A y
5 eleq1 y = B y 𝑷 B 𝑷
6 5 anbi2d y = B A 𝑷 y 𝑷 A 𝑷 B 𝑷
7 psseq2 y = B A y A B
8 6 7 anbi12d y = B A 𝑷 y 𝑷 A y A 𝑷 B 𝑷 A B
9 df-ltp < 𝑷 = x y | x 𝑷 y 𝑷 x y
10 4 8 9 brabg A 𝑷 B 𝑷 A < 𝑷 B A 𝑷 B 𝑷 A B
11 10 bianabs A 𝑷 B 𝑷 A < 𝑷 B A B