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Theorem ssrecnpr 41099
 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 4549 . 2 (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ))
2 eqimss2 3973 . . 3 (𝑆 = ℝ → ℝ ⊆ 𝑆)
3 ax-resscn 10598 . . . 4 ℝ ⊆ ℂ
4 sseq2 3942 . . . 4 (𝑆 = ℂ → (ℝ ⊆ 𝑆 ↔ ℝ ⊆ ℂ))
53, 4mpbiri 261 . . 3 (𝑆 = ℂ → ℝ ⊆ 𝑆)
62, 5jaoi 854 . 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 844   = wceq 1538   ∈ wcel 2111   ⊆ wss 3882  {cpr 4529  ℂcc 10539  ℝcr 10540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-resscn 10598 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3887  df-in 3889  df-ss 3899  df-sn 4528  df-pr 4530 This theorem is referenced by: (None)
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