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Metamath Proof Explorer

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 biimtrid ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐵 ≠ ∅ → 𝐴 ≼ ( 𝐴 × 𝐵 ) ) )
18 17 3impia ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅ ) → 𝐴 ≼ ( 𝐴 × 𝐵 ) )