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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > grurankcld | Structured version Visualization version GIF version |
Description: Grothendieck universes are closed under the rank function. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
Ref | Expression |
---|---|
grurankcld.1 | β’ (π β πΊ β Univ) |
grurankcld.2 | β’ (π β π΄ β πΊ) |
Ref | Expression |
---|---|
grurankcld | β’ (π β (rankβπ΄) β πΊ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grurankcld.2 | . . . 4 β’ (π β π΄ β πΊ) | |
2 | grurankcld.1 | . . . . 5 β’ (π β πΊ β Univ) | |
3 | 2 | elexd 3467 | . . . . . 6 β’ (π β πΊ β V) |
4 | unir1 9757 | . . . . . 6 β’ βͺ (π 1 β On) = V | |
5 | 3, 4 | eleqtrrdi 2845 | . . . . 5 β’ (π β πΊ β βͺ (π 1 β On)) |
6 | eqid 2733 | . . . . . 6 β’ (πΊ β© On) = (πΊ β© On) | |
7 | 6 | grur1 10764 | . . . . 5 β’ ((πΊ β Univ β§ πΊ β βͺ (π 1 β On)) β πΊ = (π 1β(πΊ β© On))) |
8 | 2, 5, 7 | syl2anc 585 | . . . 4 β’ (π β πΊ = (π 1β(πΊ β© On))) |
9 | 1, 8 | eleqtrd 2836 | . . 3 β’ (π β π΄ β (π 1β(πΊ β© On))) |
10 | 9 | r1rankcld 42603 | . 2 β’ (π β (rankβπ΄) β (π 1β(πΊ β© On))) |
11 | 10, 8 | eleqtrrd 2837 | 1 β’ (π β (rankβπ΄) β πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3447 β© cin 3913 βͺ cuni 4869 β cima 5640 Oncon0 6321 βcfv 6500 π 1cr1 9706 rankcrnk 9707 Univcgru 10734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-reg 9536 ax-inf2 9585 ax-ac2 10407 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-tc 9681 df-r1 9708 df-rank 9709 df-card 9883 df-cf 9885 df-acn 9886 df-ac 10060 df-wina 10628 df-ina 10629 df-gru 10735 |
This theorem is referenced by: gruscottcld 42621 |
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