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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 9789 | . . 3 β’ (rankβπ₯) β On | |
| 2 | inaex 43046 | . . 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 767 | . . . . . 6 β’ (((π§ β Inacc β§ (rankβπ₯) β π§) β§ π¦ = (π 1βπ§)) β (rankβπ₯) β π§) | |
| 5 | inawina 10684 | . . . . . . . . 9 β’ (π§ β Inacc β π§ β Inaccw) | |
| 6 | winaon 10682 | . . . . . . . . 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 724 | . . . . . . 7 β’ (((π§ β Inacc β§ (rankβπ₯) β π§) β§ π¦ = (π 1βπ§)) β π§ β On) |
| 9 | vex 3478 | . . . . . . . 8 β’ π₯ β V | |
| 10 | 9 | rankr1a 9830 | . . . . . . 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 256 | . . . . 5 β’ (((π§ β Inacc β§ (rankβπ₯) β π§) β§ π¦ = (π 1βπ§)) β π₯ β (π 1βπ§)) |
| 13 | simpr 485 | . . . . 5 β’ (((π§ β Inacc β§ (rankβπ₯) β π§) β§ π¦ = (π 1βπ§)) β π¦ = (π 1βπ§)) | |
| 14 | 12, 13 | eleqtrrd 2836 | . . . 4 β’ (((π§ β Inacc β§ (rankβπ₯) β π§) β§ π¦ = (π 1βπ§)) β π₯ β π¦) |
| 15 | simpl 483 | . . . . 5 β’ ((π§ β Inacc β§ (rankβπ₯) β π§) β π§ β Inacc) | |
| 16 | 15 | inagrud 43045 | . . . 4 β’ ((π§ β Inacc β§ (rankβπ₯) β π§) β (π 1βπ§) β Univ) |
| 17 | 14, 16 | rspcime 3616 | . . 3 β’ ((π§ β Inacc β§ (rankβπ₯) β π§) β βπ¦ β Univ π₯ β π¦) |
| 18 | 17 | rexlimiva 3147 | . 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 396 = wceq 1541 β wcel 2106 βwrex 3070 Oncon0 6364 βcfv 6543 π 1cr1 9756 rankcrnk 9757 Inaccwcwina 10676 Inacccina 10677 Univcgru 10784 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-reg 9586 ax-inf2 9635 ax-ac2 10457 ax-groth 10817 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-smo 8345 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-oi 9504 df-har 9551 df-r1 9758 df-rank 9759 df-card 9933 df-aleph 9934 df-cf 9935 df-acn 9936 df-ac 10110 df-wina 10678 df-ina 10679 df-tsk 10743 df-gru 10785 |
| This theorem is referenced by: rr-groth 43048 rr-grothprim 43049 rr-grothshort 43053 |
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