Theorem xpsneng 6876

Description: A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)

Assertion

Ref	Expression
xpsneng	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × {𝐵}) ≈ 𝐴)

Proof of Theorem xpsneng

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpeq1 4673	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 × {𝑦}) = (𝐴 × {𝑦}))
2		id 19	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
3	1, 2	breq12d 4042	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 × {𝑦}) ≈ 𝑥 ↔ (𝐴 × {𝑦}) ≈ 𝐴))
4		sneq 3629	. . . 4 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵})
5	4	xpeq2d 4683	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 × {𝑦}) = (𝐴 × {𝐵}))
6	5	breq1d 4039	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 × {𝑦}) ≈ 𝐴 ↔ (𝐴 × {𝐵}) ≈ 𝐴))
7		vex 2763	. . 3 ⊢ 𝑥 ∈ V
8		vex 2763	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	xpsnen 6875	. 2 ⊢ (𝑥 × {𝑦}) ≈ 𝑥
10	3, 6, 9	vtocl2g 2824	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × {𝐵}) ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  {csn 3618   class class class wbr 4029   × cxp 4657   ≈ cen 6792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-en 6795

This theorem is referenced by:  xp1en  6877  xpsnen2g  6883  xpdom3m  6888  hashxp  10897  pwf1oexmid  15490