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Theorem ssrecnpr 40130
 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 4488 . 2 (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ))
2 eqimss2 3940 . . 3 (𝑆 = ℝ → ℝ ⊆ 𝑆)
3 ax-resscn 10429 . . . 4 ℝ ⊆ ℂ
4 sseq2 3909 . . . 4 (𝑆 = ℂ → (ℝ ⊆ 𝑆 ↔ ℝ ⊆ ℂ))
53, 4mpbiri 259 . . 3 (𝑆 = ℂ → ℝ ⊆ 𝑆)
62, 5jaoi 852 . 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 842   = wceq 1520   ∈ wcel 2079   ⊆ wss 3854  {cpr 4468  ℂcc 10370  ℝcr 10371 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767  ax-resscn 10429 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-v 3434  df-un 3859  df-in 3861  df-ss 3869  df-sn 4467  df-pr 4469 This theorem is referenced by: (None)
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