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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 4145 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
2 | eqimss2 3621 | . . 3 ⊢ (𝑆 = ℝ → ℝ ⊆ 𝑆) | |
3 | ax-resscn 9872 | . . . 4 ⊢ ℝ ⊆ ℂ | |
4 | sseq2 3590 | . . . 4 ⊢ (𝑆 = ℂ → (ℝ ⊆ 𝑆 ↔ ℝ ⊆ ℂ)) | |
5 | 3, 4 | mpbiri 247 | . . 3 ⊢ (𝑆 = ℂ → ℝ ⊆ 𝑆) |
6 | 2, 5 | jaoi 393 | . 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 382 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 {cpr 4127 ℂcc 9813 ℝcr 9814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 df-sn 4126 df-pr 4128 |
This theorem is referenced by: (None) |
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