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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 10809 | . . 3 ⊢ (𝐼 ∈ Inacc → (𝑅1‘𝐼) ∈ Tarski) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → (𝑅1‘𝐼) ∈ Tarski) |
4 | r1tr 9807 | . 2 ⊢ Tr (𝑅1‘𝐼) | |
5 | grutsk1 10852 | . 2 ⊢ (((𝑅1‘𝐼) ∈ Tarski ∧ Tr (𝑅1‘𝐼)) → (𝑅1‘𝐼) ∈ Univ) | |
6 | 3, 4, 5 | sylancl 585 | 1 ⊢ (𝜑 → (𝑅1‘𝐼) ∈ Univ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2104 Tr wtr 5266 ‘cfv 6558 𝑅1cr1 9793 Inacccina 10714 Tarskictsk 10779 Univcgru 10821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-inf2 9672 ax-ac2 10494 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-int 4954 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-om 7881 df-1st 8007 df-2nd 8008 df-frecs 8299 df-wrecs 8330 df-smo 8379 df-recs 8404 df-rdg 8443 df-1o 8499 df-2o 8500 df-er 8738 df-map 8861 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-oi 9541 df-har 9588 df-r1 9795 df-rank 9796 df-card 9970 df-aleph 9971 df-cf 9972 df-acn 9973 df-ac 10147 df-wina 10715 df-ina 10716 df-tsk 10780 df-gru 10822 |
This theorem is referenced by: gruex 44253 |
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