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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssrecnpr | Structured version Visualization version GIF version |
Description: ℝ is a subset of both ℝ and ℂ. (Contributed by Steve Rodriguez, 22-Nov-2015.) |
Ref | Expression |
---|---|
ssrecnpr | ⊢ (𝑆 ∈ {ℝ, ℂ} → ℝ ⊆ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpri 4612 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
2 | eqimss2 4005 | . . 3 ⊢ (𝑆 = ℝ → ℝ ⊆ 𝑆) | |
3 | ax-resscn 11116 | . . . 4 ⊢ ℝ ⊆ ℂ | |
4 | sseq2 3974 | . . . 4 ⊢ (𝑆 = ℂ → (ℝ ⊆ 𝑆 ↔ ℝ ⊆ ℂ)) | |
5 | 3, 4 | mpbiri 258 | . . 3 ⊢ (𝑆 = ℂ → ℝ ⊆ 𝑆) |
6 | 2, 5 | jaoi 856 | . 2 ⊢ ((𝑆 = ℝ ∨ 𝑆 = ℂ) → ℝ ⊆ 𝑆) |
7 | 1, 6 | syl 17 | 1 ⊢ (𝑆 ∈ {ℝ, ℂ} → ℝ ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ⊆ wss 3914 {cpr 4592 ℂcc 11057 ℝcr 11058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-resscn 11116 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3449 df-un 3919 df-in 3921 df-ss 3931 df-sn 4591 df-pr 4593 |
This theorem is referenced by: (None) |
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