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Mathbox for Rohan Ridenour |
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| 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 9810 | . . 3 β’ (rankβπ₯) β On | |
| 2 | inaex 43657 | . . 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 768 | . . . . . 6 β’ (((π§ β Inacc β§ (rankβπ₯) β π§) β§ π¦ = (π 1βπ§)) β (rankβπ₯) β π§) | |
| 5 | inawina 10705 | . . . . . . . . 9 β’ (π§ β Inacc β π§ β Inaccw) | |
| 6 | winaon 10703 | . . . . . . . . 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 725 | . . . . . . 7 β’ (((π§ β Inacc β§ (rankβπ₯) β π§) β§ π¦ = (π 1βπ§)) β π§ β On) |
| 9 | vex 3473 | . . . . . . . 8 β’ π₯ β V | |
| 10 | 9 | rankr1a 9851 | . . . . . . 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 2831 | . . . 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 43656 | . . . 4 β’ ((π§ β Inacc β§ (rankβπ₯) β π§) β (π 1βπ§) β Univ) |
| 17 | 14, 16 | rspcime 3612 | . . 3 β’ ((π§ β Inacc β§ (rankβπ₯) β π§) β βπ¦ β Univ π₯ β π¦) |
| 18 | 17 | rexlimiva 3142 | . 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 1534 β wcel 2099 βwrex 3065 Oncon0 6363 βcfv 6542 π 1cr1 9777 rankcrnk 9778 Inaccwcwina 10697 Inacccina 10698 Univcgru 10805 |
| 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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-reg 9607 ax-inf2 9656 ax-ac2 10478 ax-groth 10838 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-smo 8360 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8838 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-oi 9525 df-har 9572 df-r1 9779 df-rank 9780 df-card 9954 df-aleph 9955 df-cf 9956 df-acn 9957 df-ac 10131 df-wina 10699 df-ina 10700 df-tsk 10764 df-gru 10806 |
| This theorem is referenced by: rr-groth 43659 rr-grothprim 43660 rr-grothshort 43664 |
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