Theorem ssrecnpr 41896

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

Step	Hyp	Ref	Expression
1		elpri 4589	. 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ))
2		eqimss2 3983	. . 3 ⊢ (𝑆 = ℝ → ℝ ⊆ 𝑆)
3		ax-resscn 10929	. . . 4 ⊢ ℝ ⊆ ℂ
4		sseq2 3952	. . . 4 ⊢ (𝑆 = ℂ → (ℝ ⊆ 𝑆 ↔ ℝ ⊆ ℂ))
5	3, 4	mpbiri 257	. . 3 ⊢ (𝑆 = ℂ → ℝ ⊆ 𝑆)
6	2, 5	jaoi 854	. 2 ⊢ ((𝑆 = ℝ ∨ 𝑆 = ℂ) → ℝ ⊆ 𝑆)
7	1, 6	syl 17	1 ⊢ (𝑆 ∈ {ℝ, ℂ} → ℝ ⊆ 𝑆)

Syntax hints:   → wi 4   ∨ wo 844   = wceq 1542   ∈ wcel 2110   ⊆ wss 3892  {cpr 4569  ℂcc 10870  ℝcr 10871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-resscn 10929

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-un 3897  df-in 3899  df-ss 3909  df-sn 4568  df-pr 4570

This theorem is referenced by: (None)