Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gruscottcld | Structured version Visualization version GIF version |
Description: If a Grothendieck universe contains an element of a Scott's trick set, it contains the Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
gruscottcld.1 | ⊢ (𝜑 → 𝐺 ∈ Univ) |
gruscottcld.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐺) |
gruscottcld.3 | ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴) |
Ref | Expression |
---|---|
gruscottcld | ⊢ (𝜑 → Scott 𝐴 ∈ 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gruscottcld.1 | . 2 ⊢ (𝜑 → 𝐺 ∈ Univ) | |
2 | gruscottcld.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴) | |
3 | 2 | scottrankd 40674 | . . 3 ⊢ (𝜑 → (rank‘Scott 𝐴) = suc (rank‘𝐵)) |
4 | gruscottcld.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐺) | |
5 | 1, 4 | grurankcld 40659 | . . . 4 ⊢ (𝜑 → (rank‘𝐵) ∈ 𝐺) |
6 | 1, 5 | grusucd 40656 | . . 3 ⊢ (𝜑 → suc (rank‘𝐵) ∈ 𝐺) |
7 | 3, 6 | eqeltrd 2913 | . 2 ⊢ (𝜑 → (rank‘Scott 𝐴) ∈ 𝐺) |
8 | scottex2 40671 | . . 3 ⊢ Scott 𝐴 ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → Scott 𝐴 ∈ V) |
10 | 1, 7, 9 | grurankrcld 40660 | 1 ⊢ (𝜑 → Scott 𝐴 ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3486 suc csuc 6179 ‘cfv 6341 rankcrnk 9178 Univcgru 10198 Scott cscott 40661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-reg 9042 ax-inf2 9090 ax-ac2 9871 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-er 8275 df-map 8394 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-tc 9165 df-r1 9179 df-rank 9180 df-card 9354 df-cf 9356 df-acn 9357 df-ac 9528 df-wina 10092 df-ina 10093 df-gru 10199 df-scott 40662 |
This theorem is referenced by: grucollcld 40686 |
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