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Theorem ixpprc 8960
Description: A cartesian product of proper-class many sets is empty, because any function in the cartesian product has to be a set with domain 𝐴, which is not possible for a proper class domain. (Contributed by Mario Carneiro, 25-Jan-2015.)
Assertion
Ref Expression
ixpprc 𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem ixpprc
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 neq0 4351 . . 3 X𝑥𝐴 𝐵 = ∅ ↔ ∃𝑓 𝑓X𝑥𝐴 𝐵)
2 ixpfn 8944 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓 Fn 𝐴)
3 fndm 6670 . . . . . 6 (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴)
4 vex 3483 . . . . . . 7 𝑓 ∈ V
54dmex 7932 . . . . . 6 dom 𝑓 ∈ V
63, 5eqeltrrdi 2849 . . . . 5 (𝑓 Fn 𝐴𝐴 ∈ V)
72, 6syl 17 . . . 4 (𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
87exlimiv 1929 . . 3 (∃𝑓 𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
91, 8sylbi 217 . 2 X𝑥𝐴 𝐵 = ∅ → 𝐴 ∈ V)
109con1i 147 1 𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1539  wex 1778  wcel 2107  Vcvv 3479  c0 4332  dom cdm 5684   Fn wfn 6555  Xcixp 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-ixp 8939
This theorem is referenced by:  ixpexg  8963  ixpssmap2g  8968  ixpssmapg  8969  resixpfo  8977
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