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Theorem ixpprc 7971
 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 3963 . . 3 X𝑥𝐴 𝐵 = ∅ ↔ ∃𝑓 𝑓X𝑥𝐴 𝐵)
2 ixpfn 7956 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓 Fn 𝐴)
3 fndm 6028 . . . . . 6 (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴)
4 vex 3234 . . . . . . 7 𝑓 ∈ V
54dmex 7141 . . . . . 6 dom 𝑓 ∈ V
63, 5syl6eqelr 2739 . . . . 5 (𝑓 Fn 𝐴𝐴 ∈ V)
72, 6syl 17 . . . 4 (𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
87exlimiv 1898 . . 3 (∃𝑓 𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
91, 8sylbi 207 . 2 X𝑥𝐴 𝐵 = ∅ → 𝐴 ∈ V)
109con1i 144 1 𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1523  ∃wex 1744   ∈ wcel 2030  Vcvv 3231  ∅c0 3948  dom cdm 5143   Fn wfn 5921  Xcixp 7950 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-ixp 7951 This theorem is referenced by:  ixpexg  7974  ixpssmap2g  7979  ixpssmapg  7980  resixpfo  7988
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