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Theorem dprddomprc 18607
Description: A family of subgroups indexed by a proper class cannot be a family of subgroups for an internal direct product. (Contributed by AV, 13-Jul-2019.)
Assertion
Ref Expression
dprddomprc (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆)

Proof of Theorem dprddomprc
StepHypRef Expression
1 df-nel 3047 . . 3 (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V)
2 dmexg 7248 . . . 4 (𝑆 ∈ V → dom 𝑆 ∈ V)
32con3i 151 . . 3 (¬ dom 𝑆 ∈ V → ¬ 𝑆 ∈ V)
41, 3sylbi 207 . 2 (dom 𝑆 ∉ V → ¬ 𝑆 ∈ V)
5 reldmdprd 18604 . . 3 Rel dom DProd
65brrelex2i 5298 . 2 (𝐺dom DProd 𝑆𝑆 ∈ V)
74, 6nsyl 137 1 (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2145  wnel 3046  Vcvv 3351   class class class wbr 4787  dom cdm 5250   DProd cdprd 18600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-xp 5256  df-rel 5257  df-cnv 5258  df-dm 5260  df-rn 5261  df-oprab 6800  df-mpt2 6801  df-dprd 18602
This theorem is referenced by:  dprddomcld  18608  dprdsubg  18631
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