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Theorem ixpprc 8163
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 4128 . . 3 X𝑥𝐴 𝐵 = ∅ ↔ ∃𝑓 𝑓X𝑥𝐴 𝐵)
2 ixpfn 8148 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓 Fn 𝐴)
3 fndm 6198 . . . . . 6 (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴)
4 vex 3393 . . . . . . 7 𝑓 ∈ V
54dmex 7326 . . . . . 6 dom 𝑓 ∈ V
63, 5syl6eqelr 2893 . . . . 5 (𝑓 Fn 𝐴𝐴 ∈ V)
72, 6syl 17 . . . 4 (𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
87exlimiv 2023 . . 3 (∃𝑓 𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
91, 8sylbi 208 . 2 X𝑥𝐴 𝐵 = ∅ → 𝐴 ∈ V)
109con1i 146 1 𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1637  wex 1859  wcel 2158  Vcvv 3390  c0 4113  dom cdm 5308   Fn wfn 6093  Xcixp 8142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ral 3100  df-rex 3101  df-rab 3104  df-v 3392  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4627  df-br 4841  df-opab 4903  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-iota 6061  df-fun 6100  df-fn 6101  df-fv 6106  df-ixp 8143
This theorem is referenced by:  ixpexg  8166  ixpssmap2g  8171  ixpssmapg  8172  resixpfo  8180
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