Nearby theorems
|
|
Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > gruex | Structured version Visualization version GIF version | ||
| Description: Assuming the Tarski-Grothendieck axiom, every set is contained in a Grothendieck universe. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
| Ref | Expression |
|---|---|
| gruex | β’ βπ¦ β Univ π₯ β π¦ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rankon 9786 | . . 3 β’ (rankβπ₯) β On | |
| 2 | inaex 43545 | . . 3 β’ ((rankβπ₯) β On β βπ§ β Inacc (rankβπ₯) β π§) | |
| 3 | 1, 2 | ax-mp 5 | . 2 β’ βπ§ β Inacc (rankβπ₯) β π§ |
| 4 | simplr 766 | . . . . . 6 β’ (((π§ β Inacc β§ (rankβπ₯) β π§) β§ π¦ = (π 1βπ§)) β (rankβπ₯) β π§) | |
| 5 | inawina 10681 | . . . . . . . . 9 β’ (π§ β Inacc β π§ β Inaccw) | |
| 6 | winaon 10679 | . . . . . . . . 9 β’ (π§ β Inaccw β π§ β On) | |
| 7 | 5, 6 | syl 17 | . . . . . . . 8 β’ (π§ β Inacc β π§ β On) |
| 8 | 7 | ad2antrr 723 | . . . . . . 7 β’ (((π§ β Inacc β§ (rankβπ₯) β π§) β§ π¦ = (π 1βπ§)) β π§ β On) |
| 9 | vex 3470 | . . . . . . . 8 β’ π₯ β V | |
| 10 | 9 | rankr1a 9827 | . . . . . . 7 β’ (π§ β On β (π₯ β (π 1βπ§) β (rankβπ₯) β π§)) |
| 11 | 8, 10 | syl 17 | . . . . . 6 β’ (((π§ β Inacc β§ (rankβπ₯) β π§) β§ π¦ = (π 1βπ§)) β (π₯ β (π 1βπ§) β (rankβπ₯) β π§)) |
| 12 | 4, 11 | mpbird 257 | . . . . 5 β’ (((π§ β Inacc β§ (rankβπ₯) β π§) β§ π¦ = (π 1βπ§)) β π₯ β (π 1βπ§)) |
| 13 | simpr 484 | . . . . 5 β’ (((π§ β Inacc β§ (rankβπ₯) β π§) β§ π¦ = (π 1βπ§)) β π¦ = (π 1βπ§)) | |
| 14 | 12, 13 | eleqtrrd 2828 | . . . 4 β’ (((π§ β Inacc β§ (rankβπ₯) β π§) β§ π¦ = (π 1βπ§)) β π₯ β π¦) |
| 15 | simpl 482 | . . . . 5 β’ ((π§ β Inacc β§ (rankβπ₯) β π§) β π§ β Inacc) | |
| 16 | 15 | inagrud 43544 | . . . 4 β’ ((π§ β Inacc β§ (rankβπ₯) β π§) β (π 1βπ§) β Univ) |
| 17 | 14, 16 | rspcime 3608 | . . 3 β’ ((π§ β Inacc β§ (rankβπ₯) β π§) β βπ¦ β Univ π₯ β π¦) |
| 18 | 17 | rexlimiva 3139 | . 2 β’ (βπ§ β Inacc (rankβπ₯) β π§ β βπ¦ β Univ π₯ β π¦) |
| 19 | 3, 18 | ax-mp 5 | 1 β’ βπ¦ β Univ π₯ β π¦ |
| Colors of variables: wff setvar class |
| Syntax hints: β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwrex 3062 Oncon0 6354 βcfv 6533 π 1cr1 9753 rankcrnk 9754 Inaccwcwina 10673 Inacccina 10674 Univcgru 10781 |
| 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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-reg 9583 ax-inf2 9632 ax-ac2 10454 ax-groth 10814 |
| 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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-smo 8341 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-oi 9501 df-har 9548 df-r1 9755 df-rank 9756 df-card 9930 df-aleph 9931 df-cf 9932 df-acn 9933 df-ac 10107 df-wina 10675 df-ina 10676 df-tsk 10740 df-gru 10782 |
| This theorem is referenced by: rr-groth 43547 rr-grothprim 43548 rr-grothshort 43552 |
| Copyright terms: Public domain | W3C validator |