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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 40940 | . 2 ⊢ (𝜑 → (𝑅1‘(rank‘𝐴)) ∈ 𝐺) |
4 | grurankrcld.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | r1rankid 9272 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) |
7 | gruss 10207 | . 2 ⊢ ((𝐺 ∈ Univ ∧ (𝑅1‘(rank‘𝐴)) ∈ 𝐺 ∧ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) → 𝐴 ∈ 𝐺) | |
8 | 1, 3, 6, 7 | syl3anc 1368 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ⊆ wss 3881 ‘cfv 6324 𝑅1cr1 9175 rankcrnk 9176 Univcgru 10201 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-map 8391 df-r1 9177 df-rank 9178 df-gru 10202 |
This theorem is referenced by: gruscottcld 40957 |
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