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Metamath Proof Explorer

Theorem xpdom1

Description: Dominance law for Cartesian product. Theorem 6L(c) of Enderton p. 149. (Contributed by NM, 28-Sep-2004) (Revised by NM, 29-Mar-2006) (Revised by Mario Carneiro, 7-May-2015)

		Ref	Expression
	Hypothesis	xpdom1.2	⊢ 𝐶 ∈ V
	Assertion	xpdom1	⊢ ( 𝐴 ≼ 𝐵 → ( 𝐴 × 𝐶 ) ≼ ( 𝐵 × 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		xpdom1.2	⊢ 𝐶 ∈ V
2		xpdom1g	⊢ ( ( 𝐶 ∈ V ∧ 𝐴 ≼ 𝐵 ) → ( 𝐴 × 𝐶 ) ≼ ( 𝐵 × 𝐶 ) )
3	1 2	mpan	⊢ ( 𝐴 ≼ 𝐵 → ( 𝐴 × 𝐶 ) ≼ ( 𝐵 × 𝐶 ) )