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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 3121 | . . 3 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V) | |
2 | dmexg 7602 | . . . 4 ⊢ (𝑆 ∈ V → dom 𝑆 ∈ V) | |
3 | 2 | con3i 157 | . . 3 ⊢ (¬ dom 𝑆 ∈ V → ¬ 𝑆 ∈ V) |
4 | 1, 3 | sylbi 218 | . 2 ⊢ (dom 𝑆 ∉ V → ¬ 𝑆 ∈ V) |
5 | reldmdprd 19048 | . . 3 ⊢ Rel dom DProd | |
6 | 5 | brrelex2i 5602 | . 2 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V) |
7 | 4, 6 | nsyl 142 | 1 ⊢ (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2105 ∉ wnel 3120 Vcvv 3492 class class class wbr 5057 dom cdm 5548 DProd cdprd 19044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-oprab 7149 df-mpo 7150 df-dprd 19046 |
This theorem is referenced by: dprddomcld 19052 dprdsubg 19075 |
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