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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 4582 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
2 | eqimss2 4023 | . . 3 ⊢ (𝑆 = ℝ → ℝ ⊆ 𝑆) | |
3 | ax-resscn 10588 | . . . 4 ⊢ ℝ ⊆ ℂ | |
4 | sseq2 3992 | . . . 4 ⊢ (𝑆 = ℂ → (ℝ ⊆ 𝑆 ↔ ℝ ⊆ ℂ)) | |
5 | 3, 4 | mpbiri 260 | . . 3 ⊢ (𝑆 = ℂ → ℝ ⊆ 𝑆) |
6 | 2, 5 | jaoi 853 | . 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 843 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 {cpr 4562 ℂcc 10529 ℝcr 10530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3940 df-in 3942 df-ss 3951 df-sn 4561 df-pr 4563 |
This theorem is referenced by: (None) |
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