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Theorem xpsneng 6232
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.
StepHypRef Expression
1 xpeq1 4302 . . 3 (x = A → (x × {y}) = (A × {y}))
2 id 19 . . 3 (x = Ax = A)
31, 2breq12d 3768 . 2 (x = A → ((x × {y}) ≈ x ↔ (A × {y}) ≈ A))
4 sneq 3378 . . . 4 (y = B → {y} = {B})
54xpeq2d 4312 . . 3 (y = B → (A × {y}) = (A × {B}))
65breq1d 3765 . 2 (y = B → ((A × {y}) ≈ A ↔ (A × {B}) ≈ A))
7 vex 2554 . . 3 x V
8 vex 2554 . . 3 y V
97, 8xpsnen 6231 . 2 (x × {y}) ≈ x
103, 6, 9vtocl2g 2611 1 ((A 𝑉 B 𝑊) → (A × {B}) ≈ A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  {csn 3367   class class class wbr 3755   × cxp 4286  cen 6155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-en 6158
This theorem is referenced by:  xp1en  6233  xpsnen2g  6239  xpdom3m  6244
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