Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpsneng Structured version   Visualization version   GIF version

Theorem xpsneng 8588
 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 5534 . . 3 (𝑥 = 𝐴 → (𝑥 × {𝑦}) = (𝐴 × {𝑦}))
2 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
31, 2breq12d 5044 . 2 (𝑥 = 𝐴 → ((𝑥 × {𝑦}) ≈ 𝑥 ↔ (𝐴 × {𝑦}) ≈ 𝐴))
4 sneq 4535 . . . 4 (𝑦 = 𝐵 → {𝑦} = {𝐵})
54xpeq2d 5550 . . 3 (𝑦 = 𝐵 → (𝐴 × {𝑦}) = (𝐴 × {𝐵}))
65breq1d 5041 . 2 (𝑦 = 𝐵 → ((𝐴 × {𝑦}) ≈ 𝐴 ↔ (𝐴 × {𝐵}) ≈ 𝐴))
7 vex 3444 . . 3 𝑥 ∈ V
8 vex 3444 . . 3 𝑦 ∈ V
97, 8xpsnen 8587 . 2 (𝑥 × {𝑦}) ≈ 𝑥
103, 6, 9vtocl2g 3520 1 ((𝐴𝑉𝐵𝑊) → (𝐴 × {𝐵}) ≈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {csn 4525   class class class wbr 5031   × cxp 5518   ≈ cen 8492 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4840  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-en 8496 This theorem is referenced by:  xp1en  8589  xpsnen2g  8596  xpdom3  8601  disjen  8661  unxpdom2  8713  sucxpdom  8714  gchxpidm  10083  frlmiscvec  20543
 Copyright terms: Public domain W3C validator