![]() |
Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
|
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 3489 | . . . . . 6 β’ (π β πΊ β V) |
4 | unir1 9810 | . . . . . 6 β’ βͺ (π 1 β On) = V | |
5 | 3, 4 | eleqtrrdi 2838 | . . . . 5 β’ (π β πΊ β βͺ (π 1 β On)) |
6 | eqid 2726 | . . . . . 6 β’ (πΊ β© On) = (πΊ β© On) | |
7 | 6 | grur1 10817 | . . . . 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 583 | . . . 4 β’ (π β πΊ = (π 1β(πΊ β© On))) |
9 | 1, 8 | eleqtrd 2829 | . . 3 β’ (π β π΄ β (π 1β(πΊ β© On))) |
10 | 9 | r1rankcld 43566 | . 2 β’ (π β (rankβπ΄) β (π 1β(πΊ β© On))) |
11 | 10, 8 | eleqtrrd 2830 | 1 β’ (π β (rankβπ΄) β πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3468 β© cin 3942 βͺ cuni 4902 β cima 5672 Oncon0 6358 βcfv 6537 π 1cr1 9759 rankcrnk 9760 Univcgru 10787 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-reg 9589 ax-inf2 9638 ax-ac2 10460 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-tc 9734 df-r1 9761 df-rank 9762 df-card 9936 df-cf 9938 df-acn 9939 df-ac 10113 df-wina 10681 df-ina 10682 df-gru 10788 |
This theorem is referenced by: gruscottcld 43584 |
Copyright terms: Public domain | W3C validator |