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Theorem ssrecnpr 37529
 Description: ℝ is a subset of both ℝ and ℂ. (Contributed by Steve Rodriguez, 22-Nov-2015.)
Assertion
Ref Expression
ssrecnpr (𝑆 ∈ {ℝ, ℂ} → ℝ ⊆ 𝑆)

Proof of Theorem ssrecnpr
StepHypRef 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))
53, 4mpbiri 247 . . 3 (𝑆 = ℂ → ℝ ⊆ 𝑆)
62, 5jaoi 393 . 2 ((𝑆 = ℝ ∨ 𝑆 = ℂ) → ℝ ⊆ 𝑆)
71, 6syl 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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