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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > inagrud | Structured version Visualization version GIF version |
Description: Inaccessible levels of the cumulative hierarchy are Grothendieck universes. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
inagrud.1 | ⊢ (𝜑 → 𝐼 ∈ Inacc) |
Ref | Expression |
---|---|
inagrud | ⊢ (𝜑 → (𝑅1‘𝐼) ∈ Univ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inagrud.1 | . . 3 ⊢ (𝜑 → 𝐼 ∈ Inacc) | |
2 | inatsk 10814 | . . 3 ⊢ (𝐼 ∈ Inacc → (𝑅1‘𝐼) ∈ Tarski) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → (𝑅1‘𝐼) ∈ Tarski) |
4 | r1tr 9812 | . 2 ⊢ Tr (𝑅1‘𝐼) | |
5 | grutsk1 10857 | . 2 ⊢ (((𝑅1‘𝐼) ∈ Tarski ∧ Tr (𝑅1‘𝐼)) → (𝑅1‘𝐼) ∈ Univ) | |
6 | 3, 4, 5 | sylancl 586 | 1 ⊢ (𝜑 → (𝑅1‘𝐼) ∈ Univ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 Tr wtr 5257 ‘cfv 6559 𝑅1cr1 9798 Inacccina 10719 Tarskictsk 10784 Univcgru 10826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-inf2 9677 ax-ac2 10499 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-smo 8382 df-recs 8407 df-rdg 8446 df-1o 8502 df-2o 8503 df-er 8741 df-map 8864 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-oi 9546 df-har 9593 df-r1 9800 df-rank 9801 df-card 9975 df-aleph 9976 df-cf 9977 df-acn 9978 df-ac 10152 df-wina 10720 df-ina 10721 df-tsk 10785 df-gru 10827 |
This theorem is referenced by: gruex 44295 |
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