Description: A weak universe containing _om contains the complex number construction. This theorem is construction-dependent in the literal sense, but will also be satisfied by any other reasonable implementation of the complex numbers. (Contributed by Mario Carneiro, 2-Jan-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | wuncn.1 | ⊢ ( 𝜑 → 𝑈 ∈ WUni ) | |
wuncn.2 | ⊢ ( 𝜑 → ω ∈ 𝑈 ) | ||
Assertion | wuncn | ⊢ ( 𝜑 → ℂ ∈ 𝑈 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wuncn.1 | ⊢ ( 𝜑 → 𝑈 ∈ WUni ) | |
2 | wuncn.2 | ⊢ ( 𝜑 → ω ∈ 𝑈 ) | |
3 | df-c | ⊢ ℂ = ( R × R ) | |
4 | df-nr | ⊢ R = ( ( P × P ) / ~R ) | |
5 | df-ni | ⊢ N = ( ω ∖ { ∅ } ) | |
6 | 1 2 | wundif | ⊢ ( 𝜑 → ( ω ∖ { ∅ } ) ∈ 𝑈 ) |
7 | 5 6 | eqeltrid | ⊢ ( 𝜑 → N ∈ 𝑈 ) |
8 | 1 7 7 | wunxp | ⊢ ( 𝜑 → ( N × N ) ∈ 𝑈 ) |
9 | elpqn | ⊢ ( 𝑥 ∈ Q → 𝑥 ∈ ( N × N ) ) | |
10 | 9 | ssriv | ⊢ Q ⊆ ( N × N ) |
11 | 10 | a1i | ⊢ ( 𝜑 → Q ⊆ ( N × N ) ) |
12 | 1 8 11 | wunss | ⊢ ( 𝜑 → Q ∈ 𝑈 ) |
13 | 1 12 | wunpw | ⊢ ( 𝜑 → 𝒫 Q ∈ 𝑈 ) |
14 | prpssnq | ⊢ ( 𝑥 ∈ P → 𝑥 ⊊ Q ) | |
15 | 14 | pssssd | ⊢ ( 𝑥 ∈ P → 𝑥 ⊆ Q ) |
16 | velpw | ⊢ ( 𝑥 ∈ 𝒫 Q ↔ 𝑥 ⊆ Q ) | |
17 | 15 16 | sylibr | ⊢ ( 𝑥 ∈ P → 𝑥 ∈ 𝒫 Q ) |
18 | 17 | ssriv | ⊢ P ⊆ 𝒫 Q |
19 | 18 | a1i | ⊢ ( 𝜑 → P ⊆ 𝒫 Q ) |
20 | 1 13 19 | wunss | ⊢ ( 𝜑 → P ∈ 𝑈 ) |
21 | 1 20 20 | wunxp | ⊢ ( 𝜑 → ( P × P ) ∈ 𝑈 ) |
22 | 1 21 | wunpw | ⊢ ( 𝜑 → 𝒫 ( P × P ) ∈ 𝑈 ) |
23 | enrer | ⊢ ~R Er ( P × P ) | |
24 | 23 | a1i | ⊢ ( 𝜑 → ~R Er ( P × P ) ) |
25 | 24 | qsss | ⊢ ( 𝜑 → ( ( P × P ) / ~R ) ⊆ 𝒫 ( P × P ) ) |
26 | 1 22 25 | wunss | ⊢ ( 𝜑 → ( ( P × P ) / ~R ) ∈ 𝑈 ) |
27 | 4 26 | eqeltrid | ⊢ ( 𝜑 → R ∈ 𝑈 ) |
28 | 1 27 27 | wunxp | ⊢ ( 𝜑 → ( R × R ) ∈ 𝑈 ) |
29 | 3 28 | eqeltrid | ⊢ ( 𝜑 → ℂ ∈ 𝑈 ) |