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Mirrors > Home > MPE Home > Th. List > dprddomprc | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
dprddomprc | ⊢ (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3037 | . . 3 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V) | |
2 | dmexg 7914 | . . . 4 ⊢ (𝑆 ∈ V → dom 𝑆 ∈ V) | |
3 | 2 | con3i 154 | . . 3 ⊢ (¬ dom 𝑆 ∈ V → ¬ 𝑆 ∈ V) |
4 | 1, 3 | sylbi 216 | . 2 ⊢ (dom 𝑆 ∉ V → ¬ 𝑆 ∈ V) |
5 | reldmdprd 19997 | . . 3 ⊢ Rel dom DProd | |
6 | 5 | brrelex2i 5739 | . 2 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V) |
7 | 4, 6 | nsyl 140 | 1 ⊢ (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2099 ∉ wnel 3036 Vcvv 3462 class class class wbr 5153 dom cdm 5682 DProd cdprd 19993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 df-oprab 7428 df-mpo 7429 df-dprd 19995 |
This theorem is referenced by: dprddomcld 20001 dprdsubg 20024 |
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