Theorem xpdom3 8058

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.)

Assertion

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

Proof of Theorem xpdom3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3931	. . 3 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐵)
2		xpsneng 8045	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (𝐴 × {𝑥}) ≈ 𝐴)
3	2	3adant2 1080	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝐴 × {𝑥}) ≈ 𝐴)
4	3	ensymd 8007	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝐴 ≈ (𝐴 × {𝑥}))
5		xpexg 6960	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V)
6	5	3adant3 1081	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝐴 × 𝐵) ∈ V)
7		simp3 1063	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8	7	snssd 4340	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → {𝑥} ⊆ 𝐵)
9		xpss2 5229	. . . . . . . 8 ⊢ ({𝑥} ⊆ 𝐵 → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵))
10	8, 9	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵))
11		ssdomg 8001	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ∈ V → ((𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)))
12	6, 10, 11	sylc 65	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵))
13		endomtr 8014	. . . . . 6 ⊢ ((𝐴 ≈ (𝐴 × {𝑥}) ∧ (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)) → 𝐴 ≼ (𝐴 × 𝐵))
14	4, 12, 13	syl2anc 693	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
15	14	3expia 1267	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐵 → 𝐴 ≼ (𝐴 × 𝐵)))
16	15	exlimdv 1861	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 ∈ 𝐵 → 𝐴 ≼ (𝐴 × 𝐵)))
17	1, 16	syl5bi 232	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ≠ ∅ → 𝐴 ≼ (𝐴 × 𝐵)))
18	17	3impia 1261	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653   × cxp 5112   ≈ cen 7952   ≼ cdom 7953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-er 7742  df-en 7956  df-dom 7957

This theorem is referenced by:  mapdom2  8131  xpfir  8182  infxpabs  9034