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Theorem ixpprc 6613
 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 ixpfn 6598 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓 Fn 𝐴)
2 fndm 5222 . . . . . 6 (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴)
3 vex 2689 . . . . . . 7 𝑓 ∈ V
43dmex 4805 . . . . . 6 dom 𝑓 ∈ V
52, 4eqeltrrdi 2231 . . . . 5 (𝑓 Fn 𝐴𝐴 ∈ V)
61, 5syl 14 . . . 4 (𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
76exlimiv 1577 . . 3 (∃𝑓 𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
87con3i 621 . 2 𝐴 ∈ V → ¬ ∃𝑓 𝑓X𝑥𝐴 𝐵)
9 notm0 3383 . 2 (¬ ∃𝑓 𝑓X𝑥𝐴 𝐵X𝑥𝐴 𝐵 = ∅)
108, 9sylib 121 1 𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686  ∅c0 3363  dom cdm 4539   Fn wfn 5118  Xcixp 6592 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-ixp 6593 This theorem is referenced by: (None)
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