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Theorem xpsn 6891
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 6889 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
41, 2, 3mp2an 691 1 ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2115  Vcvv 3480  {csn 4549  cop 4555   × cxp 5540
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350
This theorem is referenced by:  dfmpt  6894  fpar  7801  mapsnconst  8446  ixpsnf1o  8492  dju1dif  9590  infdju1  9607  s1co  14191  mat1f1o  21080  txdis  22233  pt1hmeo  22407  utop2nei  22852  utop3cls  22853  imasdsf1olem  22976  ex-xp  28217  poimirlem3  34970  poimirlem4  34971  poimirlem9  34976  poimirlem28  34995  grposnOLD  35230  dib0  38370
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