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Theorem xpsneng 6684
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 ((𝐴𝑉𝐵𝑊) → (𝐴 × {𝐵}) ≈ 𝐴)

Proof of Theorem xpsneng
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpeq1 4523 . . 3 (𝑥 = 𝐴 → (𝑥 × {𝑦}) = (𝐴 × {𝑦}))
2 id 19 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
31, 2breq12d 3912 . 2 (𝑥 = 𝐴 → ((𝑥 × {𝑦}) ≈ 𝑥 ↔ (𝐴 × {𝑦}) ≈ 𝐴))
4 sneq 3508 . . . 4 (𝑦 = 𝐵 → {𝑦} = {𝐵})
54xpeq2d 4533 . . 3 (𝑦 = 𝐵 → (𝐴 × {𝑦}) = (𝐴 × {𝐵}))
65breq1d 3909 . 2 (𝑦 = 𝐵 → ((𝐴 × {𝑦}) ≈ 𝐴 ↔ (𝐴 × {𝐵}) ≈ 𝐴))
7 vex 2663 . . 3 𝑥 ∈ V
8 vex 2663 . . 3 𝑦 ∈ V
97, 8xpsnen 6683 . 2 (𝑥 × {𝑦}) ≈ 𝑥
103, 6, 9vtocl2g 2724 1 ((𝐴𝑉𝐵𝑊) → (𝐴 × {𝐵}) ≈ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  {csn 3497   class class class wbr 3899   × cxp 4507  cen 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-en 6603
This theorem is referenced by:  xp1en  6685  xpsnen2g  6691  xpdom3m  6696  hashxp  10540  pwf1oexmid  13121
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