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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > grurankrcld | Structured version Visualization version GIF version |
Description: If a Grothendieck universe contains a set's rank, it contains that set. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
Ref | Expression |
---|---|
grurankrcld.1 | ⊢ (𝜑 → 𝐺 ∈ Univ) |
grurankrcld.2 | ⊢ (𝜑 → (rank‘𝐴) ∈ 𝐺) |
grurankrcld.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
grurankrcld | ⊢ (𝜑 → 𝐴 ∈ 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grurankrcld.1 | . 2 ⊢ (𝜑 → 𝐺 ∈ Univ) | |
2 | grurankrcld.2 | . . 3 ⊢ (𝜑 → (rank‘𝐴) ∈ 𝐺) | |
3 | 1, 2 | grur1cld 42454 | . 2 ⊢ (𝜑 → (𝑅1‘(rank‘𝐴)) ∈ 𝐺) |
4 | grurankrcld.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | r1rankid 9791 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) |
7 | gruss 10728 | . 2 ⊢ ((𝐺 ∈ Univ ∧ (𝑅1‘(rank‘𝐴)) ∈ 𝐺 ∧ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) → 𝐴 ∈ 𝐺) | |
8 | 1, 3, 6, 7 | syl3anc 1371 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ⊆ wss 3908 ‘cfv 6493 𝑅1cr1 9694 rankcrnk 9695 Univcgru 10722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-reg 9524 ax-inf2 9573 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-map 8763 df-r1 9696 df-rank 9697 df-gru 10723 |
This theorem is referenced by: gruscottcld 42471 |
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