Metamath Proof Explorer


Theorem elreal2

Description: Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013) (New usage is discouraged.)

Ref Expression
Assertion elreal2 A 1 st A 𝑹 A = 1 st A 0 𝑹

Proof

Step Hyp Ref Expression
1 df-r = 𝑹 × 0 𝑹
2 1 eleq2i A A 𝑹 × 0 𝑹
3 xp1st A 𝑹 × 0 𝑹 1 st A 𝑹
4 1st2nd2 A 𝑹 × 0 𝑹 A = 1 st A 2 nd A
5 xp2nd A 𝑹 × 0 𝑹 2 nd A 0 𝑹
6 elsni 2 nd A 0 𝑹 2 nd A = 0 𝑹
7 5 6 syl A 𝑹 × 0 𝑹 2 nd A = 0 𝑹
8 7 opeq2d A 𝑹 × 0 𝑹 1 st A 2 nd A = 1 st A 0 𝑹
9 4 8 eqtrd A 𝑹 × 0 𝑹 A = 1 st A 0 𝑹
10 3 9 jca A 𝑹 × 0 𝑹 1 st A 𝑹 A = 1 st A 0 𝑹
11 eleq1 A = 1 st A 0 𝑹 A 𝑹 × 0 𝑹 1 st A 0 𝑹 𝑹 × 0 𝑹
12 0r 0 𝑹 𝑹
13 12 elexi 0 𝑹 V
14 13 snid 0 𝑹 0 𝑹
15 opelxp 1 st A 0 𝑹 𝑹 × 0 𝑹 1 st A 𝑹 0 𝑹 0 𝑹
16 14 15 mpbiran2 1 st A 0 𝑹 𝑹 × 0 𝑹 1 st A 𝑹
17 11 16 bitrdi A = 1 st A 0 𝑹 A 𝑹 × 0 𝑹 1 st A 𝑹
18 17 biimparc 1 st A 𝑹 A = 1 st A 0 𝑹 A 𝑹 × 0 𝑹
19 10 18 impbii A 𝑹 × 0 𝑹 1 st A 𝑹 A = 1 st A 0 𝑹
20 2 19 bitri A 1 st A 𝑹 A = 1 st A 0 𝑹