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Theorem ixpprc 8470
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 4262 . . 3 X𝑥𝐴 𝐵 = ∅ ↔ ∃𝑓 𝑓X𝑥𝐴 𝐵)
2 ixpfn 8454 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓 Fn 𝐴)
3 fndm 6429 . . . . . 6 (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴)
4 vex 3447 . . . . . . 7 𝑓 ∈ V
54dmex 7602 . . . . . 6 dom 𝑓 ∈ V
63, 5eqeltrrdi 2902 . . . . 5 (𝑓 Fn 𝐴𝐴 ∈ V)
72, 6syl 17 . . . 4 (𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
87exlimiv 1931 . . 3 (∃𝑓 𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
91, 8sylbi 220 . 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 1538  wex 1781  wcel 2112  Vcvv 3444  c0 4246  dom cdm 5523   Fn wfn 6323  Xcixp 8448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-ixp 8449
This theorem is referenced by:  ixpexg  8473  ixpssmap2g  8478  ixpssmapg  8479  resixpfo  8487
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