Theorem xpdom3

Description: A set is dominated by its Cartesian product with a nonempty set. Exercise 6 of Suppes p. 98. (Contributed by NM, 27-Jul-2004) (Revised by Mario Carneiro, 29-Apr-2015)

	Ref	Expression
Assertion	xpdom3	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅ ) → 𝐴 ≼ ( 𝐴 × 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		n0	⊢ ( 𝐵 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐵 )
2		xpsneng	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐴 × { 𝑥 } ) ≈ 𝐴 )
3	2	3adant2	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐴 × { 𝑥 } ) ≈ 𝐴 )
4	3	ensymd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵 ) → 𝐴 ≈ ( 𝐴 × { 𝑥 } ) )
5		xpexg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 × 𝐵 ) ∈ V )
6	5	3adant3	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐴 × 𝐵 ) ∈ V )
7		simp3	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 )
8	7	snssd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵 ) → { 𝑥 } ⊆ 𝐵 )
9		xpss2	⊢ ( { 𝑥 } ⊆ 𝐵 → ( 𝐴 × { 𝑥 } ) ⊆ ( 𝐴 × 𝐵 ) )
10	8 9	syl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐴 × { 𝑥 } ) ⊆ ( 𝐴 × 𝐵 ) )
11		ssdomg	⊢ ( ( 𝐴 × 𝐵 ) ∈ V → ( ( 𝐴 × { 𝑥 } ) ⊆ ( 𝐴 × 𝐵 ) → ( 𝐴 × { 𝑥 } ) ≼ ( 𝐴 × 𝐵 ) ) )
12	6 10 11	sylc	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐴 × { 𝑥 } ) ≼ ( 𝐴 × 𝐵 ) )
13		endomtr	⊢ ( ( 𝐴 ≈ ( 𝐴 × { 𝑥 } ) ∧ ( 𝐴 × { 𝑥 } ) ≼ ( 𝐴 × 𝐵 ) ) → 𝐴 ≼ ( 𝐴 × 𝐵 ) )
14	4 12 13	syl2anc	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵 ) → 𝐴 ≼ ( 𝐴 × 𝐵 ) )
15	14	3expia	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝑥 ∈ 𝐵 → 𝐴 ≼ ( 𝐴 × 𝐵 ) ) )
16	15	exlimdv	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 𝑥 ∈ 𝐵 → 𝐴 ≼ ( 𝐴 × 𝐵 ) ) )
17	1 16	syl5bi	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐵 ≠ ∅ → 𝐴 ≼ ( 𝐴 × 𝐵 ) ) )
18	17	3impia	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅ ) → 𝐴 ≼ ( 𝐴 × 𝐵 ) )