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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 3484 | . . . . . 6 β’ (π β πΊ β V) |
4 | unir1 9836 | . . . . . 6 β’ βͺ (π 1 β On) = V | |
5 | 3, 4 | eleqtrrdi 2836 | . . . . 5 β’ (π β πΊ β βͺ (π 1 β On)) |
6 | eqid 2725 | . . . . . 6 β’ (πΊ β© On) = (πΊ β© On) | |
7 | 6 | grur1 10843 | . . . . 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 582 | . . . 4 β’ (π β πΊ = (π 1β(πΊ β© On))) |
9 | 1, 8 | eleqtrd 2827 | . . 3 β’ (π β π΄ β (π 1β(πΊ β© On))) |
10 | 9 | r1rankcld 43733 | . 2 β’ (π β (rankβπ΄) β (π 1β(πΊ β© On))) |
11 | 10, 8 | eleqtrrd 2828 | 1 β’ (π β (rankβπ΄) β πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3463 β© cin 3938 βͺ cuni 4903 β cima 5675 Oncon0 6364 βcfv 6543 π 1cr1 9785 rankcrnk 9786 Univcgru 10813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-reg 9615 ax-inf2 9664 ax-ac2 10486 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-tc 9760 df-r1 9787 df-rank 9788 df-card 9962 df-cf 9964 df-acn 9965 df-ac 10139 df-wina 10707 df-ina 10708 df-gru 10814 |
This theorem is referenced by: gruscottcld 43751 |
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