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Metamath Proof Explorer

Theorem axresscn

Description: The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn . (Contributed by NM, 1-Mar-1995) (Proof shortened by Andrew Salmon, 12-Aug-2011) (New usage is discouraged.)

Ref Expression
Assertion axresscn ℝ ⊆ ℂ

Proof

Step Hyp Ref Expression
1 0r ⊢ 0R ∈ R
2 snssi ⊢ ( 0R ∈ R → { 0R } ⊆ R )
3 xpss2 ⊢ ( { 0R } ⊆ R → ( R × { 0R } ) ⊆ ( R × R ) )
4 1 2 3 mp2b ⊢ ( R × { 0R } ) ⊆ ( R × R )
5 df-r ⊢ ℝ = ( R × { 0R } )
6 df-c ⊢ ℂ = ( R × R )
7 4 5 6 3sstr4i ⊢ ℝ ⊆ ℂ