Theorem recnprss 15790

Description: Both ℝ and ℂ are subsets of ℂ. (Contributed by Mario Carneiro, 10-Feb-2015.)

Assertion

Ref	Expression
recnprss	⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)

Proof of Theorem recnprss

Step	Hyp	Ref	Expression
1		elpri 3732	. 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ))
2		ax-resscn 8271	. . . 4 ⊢ ℝ ⊆ ℂ
3		sseq1 3271	. . . 4 ⊢ (𝑆 = ℝ → (𝑆 ⊆ ℂ ↔ ℝ ⊆ ℂ))
4	2, 3	mpbiri 168	. . 3 ⊢ (𝑆 = ℝ → 𝑆 ⊆ ℂ)
5		eqimss 3302	. . 3 ⊢ (𝑆 = ℂ → 𝑆 ⊆ ℂ)
6	4, 5	jaoi 728	. 2 ⊢ ((𝑆 = ℝ ∨ 𝑆 = ℂ) → 𝑆 ⊆ ℂ)
7	1, 6	syl 14	1 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)

This proof depends on syntax axioms:   → wi 4   ∨ wo 720   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220  {cpr 3710  ℂcc 8177  ℝcr 8178

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-resscn 8271

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716

This theorem is used by:  dvfgg  15791  dvidsslem  15796  dvconstss  15801  dvaddxx  15806  dvmulxx  15807  dviaddf  15808  dvimulf  15809  dvmptfsum  15828