Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3124 | . . 3 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V) | |
2 | dmexg 7612 | . . . 4 ⊢ (𝑆 ∈ V → dom 𝑆 ∈ V) | |
3 | 2 | con3i 157 | . . 3 ⊢ (¬ dom 𝑆 ∈ V → ¬ 𝑆 ∈ V) |
4 | 1, 3 | sylbi 219 | . 2 ⊢ (dom 𝑆 ∉ V → ¬ 𝑆 ∈ V) |
5 | reldmdprd 19118 | . . 3 ⊢ Rel dom DProd | |
6 | 5 | brrelex2i 5608 | . 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 2110 ∉ wnel 3123 Vcvv 3494 class class class wbr 5065 dom cdm 5554 DProd cdprd 19114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-cnv 5562 df-dm 5564 df-rn 5565 df-oprab 7159 df-mpo 7160 df-dprd 19116 |
This theorem is referenced by: dprddomcld 19122 dprdsubg 19145 |
Copyright terms: Public domain | W3C validator |