ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpex GIF version

Theorem xpex 4480
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 4479 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 × 𝐵) ∈ V)
41, 2, 3mp2an 410 1 (𝐴 × 𝐵) ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 1409  Vcvv 2574   × cxp 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-opab 3846  df-xp 4378
This theorem is referenced by:  oprabex  5782  oprabex3  5783  xpsnen  6325  endisj  6328  xpcomen  6331  xpassen  6334  enqex  6515  nqex  6518  enq0ex  6594  nq0ex  6595  npex  6628  enrex  6879  addvalex  6977  axcnex  6992  ixxex  8868  shftfval  9643
  Copyright terms: Public domain W3C validator