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Mirrors > Home > ILE Home > Th. List > recnprss | GIF version |
Description: Both ℝ and ℂ are subsets of ℂ. (Contributed by Mario Carneiro, 10-Feb-2015.) |
Ref | Expression |
---|---|
recnprss | ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpri 3550 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
2 | ax-resscn 7712 | . . . 4 ⊢ ℝ ⊆ ℂ | |
3 | sseq1 3120 | . . . 4 ⊢ (𝑆 = ℝ → (𝑆 ⊆ ℂ ↔ ℝ ⊆ ℂ)) | |
4 | 2, 3 | mpbiri 167 | . . 3 ⊢ (𝑆 = ℝ → 𝑆 ⊆ ℂ) |
5 | eqimss 3151 | . . 3 ⊢ (𝑆 = ℂ → 𝑆 ⊆ ℂ) | |
6 | 4, 5 | jaoi 705 | . 2 ⊢ ((𝑆 = ℝ ∨ 𝑆 = ℂ) → 𝑆 ⊆ ℂ) |
7 | 1, 6 | syl 14 | 1 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ⊆ wss 3071 {cpr 3528 ℂcc 7618 ℝcr 7619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 |
This theorem is referenced by: dvfgg 12826 dvaddxx 12836 dvmulxx 12837 dviaddf 12838 dvimulf 12839 |
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