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Theorem xpsn 6672
Description: The Cartesian product of two singletons is the singleton consisting in the associated ordered pair. (Contributed by NM, 4-Nov-2006.)
Hypotheses
Ref Expression
xpsn.1 𝐴 ∈ V
xpsn.2 𝐵 ∈ V
Assertion
Ref Expression
xpsn ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩}

Proof of Theorem xpsn
StepHypRef Expression
1 xpsn.1 . 2 𝐴 ∈ V
2 xpsn.2 . 2 𝐵 ∈ V
3 xpsng 6671 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
41, 2, 3mp2an 682 1 ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  wcel 2107  Vcvv 3398  {csn 4398  cop 4404   × cxp 5353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142
This theorem is referenced by:  dfmpt  6675  fpar  7562  mapsnconst  8189  ixpsnf1o  8234  cda1dif  9333  infcda1  9350  s1co  13984  xpsc0  16606  xpsc1  16607  mat1f1o  20689  txdis  21844  pt1hmeo  22018  utop2nei  22462  utop3cls  22463  imasdsf1olem  22586  ex-xp  27868  poimirlem3  34038  poimirlem4  34039  poimirlem9  34044  poimirlem28  34063  grposnOLD  34305  dib0  37318
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