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Theorem xpsneng 6326
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 4386 . . 3 (𝑥 = 𝐴 → (𝑥 × {𝑦}) = (𝐴 × {𝑦}))
2 id 19 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
31, 2breq12d 3804 . 2 (𝑥 = 𝐴 → ((𝑥 × {𝑦}) ≈ 𝑥 ↔ (𝐴 × {𝑦}) ≈ 𝐴))
4 sneq 3413 . . . 4 (𝑦 = 𝐵 → {𝑦} = {𝐵})
54xpeq2d 4396 . . 3 (𝑦 = 𝐵 → (𝐴 × {𝑦}) = (𝐴 × {𝐵}))
65breq1d 3801 . 2 (𝑦 = 𝐵 → ((𝐴 × {𝑦}) ≈ 𝐴 ↔ (𝐴 × {𝐵}) ≈ 𝐴))
7 vex 2577 . . 3 𝑥 ∈ V
8 vex 2577 . . 3 𝑦 ∈ V
97, 8xpsnen 6325 . 2 (𝑥 × {𝑦}) ≈ 𝑥
103, 6, 9vtocl2g 2634 1 ((𝐴𝑉𝐵𝑊) → (𝐴 × {𝐵}) ≈ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  {csn 3402   class class class wbr 3791   × cxp 4370  cen 6249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-en 6252
This theorem is referenced by:  xp1en  6327  xpsnen2g  6333  xpdom3m  6338
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