Mathbox for Steve Rodriguez |
< Previous
Next >
Nearby theorems |
||
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 4563 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
2 | eqimss2 3958 | . . 3 ⊢ (𝑆 = ℝ → ℝ ⊆ 𝑆) | |
3 | ax-resscn 10786 | . . . 4 ⊢ ℝ ⊆ ℂ | |
4 | sseq2 3927 | . . . 4 ⊢ (𝑆 = ℂ → (ℝ ⊆ 𝑆 ↔ ℝ ⊆ ℂ)) | |
5 | 3, 4 | mpbiri 261 | . . 3 ⊢ (𝑆 = ℂ → ℝ ⊆ 𝑆) |
6 | 2, 5 | jaoi 857 | . 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 847 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 {cpr 4543 ℂcc 10727 ℝcr 10728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-resscn 10786 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-un 3871 df-in 3873 df-ss 3883 df-sn 4542 df-pr 4544 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |