Theorem xpsnen2g

Description: A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014)

	Ref	Expression
Assertion	xpsnen2g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( { 𝐴 } × 𝐵 ) ≈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		snex	⊢ { 𝐴 } ∈ V
2		xpcomeng	⊢ ( ( { 𝐴 } ∈ V ∧ 𝐵 ∈ 𝑊 ) → ( { 𝐴 } × 𝐵 ) ≈ ( 𝐵 × { 𝐴 } ) )
3	1 2	mpan	⊢ ( 𝐵 ∈ 𝑊 → ( { 𝐴 } × 𝐵 ) ≈ ( 𝐵 × { 𝐴 } ) )
4		xpsneng	⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ) → ( 𝐵 × { 𝐴 } ) ≈ 𝐵 )
5	4	ancoms	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐵 × { 𝐴 } ) ≈ 𝐵 )
6		entr	⊢ ( ( ( { 𝐴 } × 𝐵 ) ≈ ( 𝐵 × { 𝐴 } ) ∧ ( 𝐵 × { 𝐴 } ) ≈ 𝐵 ) → ( { 𝐴 } × 𝐵 ) ≈ 𝐵 )
7	3 5 6	syl2an2	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( { 𝐴 } × 𝐵 ) ≈ 𝐵 )