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Theorem ixpprc 8471
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 4306 . . 3 X𝑥𝐴 𝐵 = ∅ ↔ ∃𝑓 𝑓X𝑥𝐴 𝐵)
2 ixpfn 8455 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓 Fn 𝐴)
3 fndm 6448 . . . . . 6 (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴)
4 vex 3495 . . . . . . 7 𝑓 ∈ V
54dmex 7605 . . . . . 6 dom 𝑓 ∈ V
63, 5syl6eqelr 2919 . . . . 5 (𝑓 Fn 𝐴𝐴 ∈ V)
72, 6syl 17 . . . 4 (𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
87exlimiv 1922 . . 3 (∃𝑓 𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
91, 8sylbi 218 . 2 X𝑥𝐴 𝐵 = ∅ → 𝐴 ∈ V)
109con1i 149 1 𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  c0 4288  dom cdm 5548   Fn wfn 6343  Xcixp 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-ixp 8450
This theorem is referenced by:  ixpexg  8474  ixpssmap2g  8479  ixpssmapg  8480  resixpfo  8488
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