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Theorem ssrecnpr 42680
Description: is a subset of both and . (Contributed by Steve Rodriguez, 22-Nov-2015.)
Assertion
Ref Expression
ssrecnpr (𝑆 ∈ {ℝ, ℂ} → ℝ ⊆ 𝑆)

Proof of Theorem ssrecnpr
StepHypRef Expression
1 elpri 4612 . 2 (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ))
2 eqimss2 4005 . . 3 (𝑆 = ℝ → ℝ ⊆ 𝑆)
3 ax-resscn 11116 . . . 4 ℝ ⊆ ℂ
4 sseq2 3974 . . . 4 (𝑆 = ℂ → (ℝ ⊆ 𝑆 ↔ ℝ ⊆ ℂ))
53, 4mpbiri 258 . . 3 (𝑆 = ℂ → ℝ ⊆ 𝑆)
62, 5jaoi 856 . 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 846   = wceq 1542  wcel 2107  wss 3914  {cpr 4592  cc 11057  cr 11058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-resscn 11116
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-un 3919  df-in 3921  df-ss 3931  df-sn 4591  df-pr 4593
This theorem is referenced by: (None)
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