Theorem xpsneng 4442

Description: A set is equinumerous to its cross-product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254.

Assertion

Ref	Expression
xpsneng	⊢ ((A ∈ C ⋀ B ∈ D) → (A × {B}) ≈ A)

Proof of Theorem xpsneng

Step	Hyp	Ref	Expression
1		xpeq1 3206	. . 3 ⊢ (x = A → (x × {y}) = (A × {y}))
2		id 59	. . 3 ⊢ (x = A → x = A)
3	1, 2	breq12d 2636	. 2 ⊢ (x = A → ((x × {y}) ≈ x ↔ (A × {y}) ≈ A))
4		sneq 2421	. . . 4 ⊢ (y = B → {y} = {B})
5		xpeq2 3207	. . . 4 ⊢ ({y} = {B} → (A × {y}) = (A × {B}))
6	4, 5	syl 10	. . 3 ⊢ (y = B → (A × {y}) = (A × {B}))
7	6	breq1d 2634	. 2 ⊢ (y = B → ((A × {y}) ≈ A ↔ (A × {B}) ≈ A))
8		visset 1816	. . 3 ⊢ x ∈ V
9		visset 1816	. . 3 ⊢ y ∈ V
10	8, 9	xpsnen 4441	. 2 ⊢ (x × {y}) ≈ x
11	3, 7, 10	vtocl2g 1853	1 ⊢ ((A ∈ C ⋀ B ∈ D) → (A × {B}) ≈ A)

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  {csn 2413   class class class wbr 2624   × cxp 3174   ≈ cen 4370

This theorem is referenced by:  cdafi 4948

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-en 4374

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