Theorem xpdom3 3347

Description: A set is dominated by its cross product with a non-empty set. Exercise 6 of [Suppes] p. 98.

Hypothesis

Ref	Expression
xpdom3.1	⊢ A ∈ V

Assertion

Ref	Expression
xpdom3	⊢ (¬ B = ∅ → A ≼ (A × B))

Proof of Theorem xpdom3

Step	Hyp	Ref	Expression
1		n0 1714	. 2 ⊢ (¬ B = ∅ ↔ ∃x x ∈ B)
2		visset 1350	. . . . 5 ⊢ x ∈ V
3	2	snss 1849	. . . 4 ⊢ (x ∈ B ↔ {x} ⊆ B)
4		ssid 1519	. . . . . 6 ⊢ A ⊆ A
5		ssxp 2487	. . . . . 6 ⊢ ((A ⊆ A ∧ {x} ⊆ B) → (A × {x}) ⊆ (A × B))
6	4, 5	mpan 518	. . . . 5 ⊢ ({x} ⊆ B → (A × {x}) ⊆ (A × B))
7		xpdom3.1	. . . . . . 7 ⊢ A ∈ V
8		snex 1859	. . . . . . 7 ⊢ {x} ∈ V
9	7, 8	xpex 2488	. . . . . 6 ⊢ (A × {x}) ∈ V
10		ssdomg 3311	. . . . . 6 ⊢ ((A × {x}) ∈ V → ((A × {x}) ⊆ (A × B) → (A × {x}) ≼ (A × B)))
11	9, 10	ax-mp 6	. . . . 5 ⊢ ((A × {x}) ⊆ (A × B) → (A × {x}) ≼ (A × B))
12	7, 2	xpsnen 3339	. . . . . . 7 ⊢ (A × {x}) ≈ A
13	7, 12	ensymi 3318	. . . . . 6 ⊢ A ≈ (A × {x})
14		endomtr 3325	. . . . . 6 ⊢ ((A ≈ (A × {x}) ∧ (A × {x}) ≼ (A × B)) → A ≼ (A × B))
15	13, 14	mpan 518	. . . . 5 ⊢ ((A × {x}) ≼ (A × B) → A ≼ (A × B))
16	6, 11, 15	3syl 21	. . . 4 ⊢ ({x} ⊆ B → A ≼ (A × B))
17	3, 16	sylbi 174	. . 3 ⊢ (x ∈ B → A ≼ (A × B))
18	17	19.23aiv 952	. 2 ⊢ (∃x x ∈ B → A ≼ (A × B))
19	1, 18	sylbi 174	1 ⊢ (¬ B = ∅ → A ≼ (A × B))

Syntax hints:  ¬ wn 1   → wi 2  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  {csn 1808   class class class wbr 2054   × cxp 2408   ≈ cen 3271   ≼ cdom 3272

This theorem is referenced by:  infxpabs 4949

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-int 1966  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-er 3200  df-en 3274  df-dom 3275

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