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Theorem xpex 4453
 Description: The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)
Hypotheses
Ref Expression
xpex.1 𝐴 ∈ V
xpex.2 𝐵 ∈ V
Assertion
Ref Expression
xpex (𝐴 × 𝐵) ∈ V

Proof of Theorem xpex
StepHypRef Expression
1 xpex.1 . 2 𝐴 ∈ V
2 xpex.2 . 2 𝐵 ∈ V
3 xpexg 4452 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 × 𝐵) ∈ V)
41, 2, 3mp2an 402 1 (𝐴 × 𝐵) ∈ V
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1393  Vcvv 2557   × cxp 4343 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-opab 3819  df-xp 4351 This theorem is referenced by:  oprabex  5755  oprabex3  5756  xpsnen  6295  endisj  6298  xpcomen  6301  xpassen  6304  enqex  6458  nqex  6461  enq0ex  6537  nq0ex  6538  npex  6571  enrex  6822  addvalex  6920  axcnex  6935  ixxex  8768  shftfval  9422
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