Theorem xpdom3 4451

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		ne0 2292	. 2 ⊢ (B ≠ ∅ ↔ ∃x x ∈ B)
2		visset 1816	. . . . 5 ⊢ x ∈ V
3	2	snss 2465	. . . 4 ⊢ (x ∈ B ↔ {x} ⊆ B)
4		ssid 2083	. . . . . 6 ⊢ A ⊆ A
5		ssxp 3262	. . . . . 6 ⊢ ((A ⊆ A ⋀ {x} ⊆ B) → (A × {x}) ⊆ (A × B))
6	4, 5	mpan 697	. . . . 5 ⊢ ({x} ⊆ B → (A × {x}) ⊆ (A × B))
7		xpdom3.1	. . . . . . 7 ⊢ A ∈ V
8		snex 2756	. . . . . . 7 ⊢ {x} ∈ V
9	7, 8	xpex 3266	. . . . . 6 ⊢ (A × {x}) ∈ V
10		ssdomg 4414	. . . . . 6 ⊢ ((A × {x}) ∈ V → ((A × {x}) ⊆ (A × B) → (A × {x}) ≼ (A × B)))
11	9, 10	ax-mp 7	. . . . 5 ⊢ ((A × {x}) ⊆ (A × B) → (A × {x}) ≼ (A × B))
12	7, 2	xpsnen 4441	. . . . . . 7 ⊢ (A × {x}) ≈ A
13	7, 12	ensymi 4419	. . . . . 6 ⊢ A ≈ (A × {x})
14		endomtr 4426	. . . . . 6 ⊢ ((A ≈ (A × {x}) ⋀ (A × {x}) ≼ (A × B)) → A ≼ (A × B))
15	13, 14	mpan 697	. . . . 5 ⊢ ((A × {x}) ≼ (A × B) → A ≼ (A × B))
16	6, 11, 15	3syl 20	. . . 4 ⊢ ({x} ⊆ B → A ≼ (A × B))
17	3, 16	sylbi 199	. . 3 ⊢ (x ∈ B → A ≼ (A × B))
18	17	19.23aiv 1297	. 2 ⊢ (∃x x ∈ B → A ≼ (A × B))
19	1, 18	sylbi 199	1 ⊢ (B ≠ ∅ → A ≼ (A × B))

Syntax hints:   → wi 3   ∈ wcel 960  ∃wex 982   ≠ wne 1588  Vcvv 1814   ⊆ wss 2050  ∅c0 2283  {csn 2413   class class class wbr 2624   × cxp 3174   ≈ cen 4370   ≼ cdom 4371

This theorem is referenced by:  infxpabs 7571

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-er 4267  df-en 4374  df-dom 4375

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