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Theorem rr-grothprimbi 43655
Description: Express "every set is contained in a Grothendieck universe" using only primitives. The right side (without the outermost universal quantifier) is proven as rr-grothprim 43660. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Assertion
Ref Expression
rr-grothprimbi (∀𝑥𝑦 ∈ Univ 𝑥𝑦 ↔ ∀𝑥 ¬ ∀𝑦(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡𝑣𝑡𝑧) → ¬ (𝑣𝑦 → ¬ 𝑣𝑤)) → ¬ ∀𝑖(𝑖𝑧 → (𝑣𝑦 → (𝑖𝑣 → (𝑣𝑓 → ¬ ∀𝑢(𝑢𝑓 → (𝑖𝑢 → ¬ ∀𝑜(𝑜𝑢 → ∀𝑠(𝑠𝑜𝑠𝑤)))))))))))))
Distinct variable groups:   𝑣,𝑢   𝑢,𝑜   𝑧,𝑣,𝑡   𝑤,𝑜,𝑠   𝑦,𝑧,𝑤,𝑣,𝑓,𝑖,𝑢

Proof of Theorem rr-grothprimbi
Dummy variables 𝑘 𝑚 𝑛 𝑞 𝑝 𝑙 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 3066 . . 3 (∃𝑦 ∈ Univ 𝑥𝑦 ↔ ∃𝑦(𝑦 ∈ Univ ∧ 𝑥𝑦))
2 ancom 460 . . . . 5 ((𝑦 ∈ Univ ∧ 𝑥𝑦) ↔ (𝑥𝑦𝑦 ∈ Univ))
3 biid 261 . . . . . 6 (𝑥𝑦𝑥𝑦)
4 grumnueq 43647 . . . . . . . . 9 Univ = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
54ismnu 43621 . . . . . . . 8 (𝑦 ∈ V → (𝑦 ∈ Univ ↔ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∀𝑓𝑤𝑦 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑦 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))))
65elv 3475 . . . . . . 7 (𝑦 ∈ Univ ↔ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∀𝑓𝑤𝑦 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑦 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
7 ismnuprim 43654 . . . . . . 7 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∀𝑓𝑤𝑦 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑦 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))) ↔ ∀𝑧(𝑧𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡𝑣𝑡𝑧) → ¬ (𝑣𝑦 → ¬ 𝑣𝑤)) → ¬ ∀𝑖(𝑖𝑧 → (𝑣𝑦 → (𝑖𝑣 → (𝑣𝑓 → ¬ ∀𝑢(𝑢𝑓 → (𝑖𝑢 → ¬ ∀𝑜(𝑜𝑢 → ∀𝑠(𝑠𝑜𝑠𝑤))))))))))))
86, 7bitri 275 . . . . . 6 (𝑦 ∈ Univ ↔ ∀𝑧(𝑧𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡𝑣𝑡𝑧) → ¬ (𝑣𝑦 → ¬ 𝑣𝑤)) → ¬ ∀𝑖(𝑖𝑧 → (𝑣𝑦 → (𝑖𝑣 → (𝑣𝑓 → ¬ ∀𝑢(𝑢𝑓 → (𝑖𝑢 → ¬ ∀𝑜(𝑜𝑢 → ∀𝑠(𝑠𝑜𝑠𝑤))))))))))))
93, 8expandan 43648 . . . . 5 ((𝑥𝑦𝑦 ∈ Univ) ↔ ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡𝑣𝑡𝑧) → ¬ (𝑣𝑦 → ¬ 𝑣𝑤)) → ¬ ∀𝑖(𝑖𝑧 → (𝑣𝑦 → (𝑖𝑣 → (𝑣𝑓 → ¬ ∀𝑢(𝑢𝑓 → (𝑖𝑢 → ¬ ∀𝑜(𝑜𝑢 → ∀𝑠(𝑠𝑜𝑠𝑤)))))))))))))
102, 9bitri 275 . . . 4 ((𝑦 ∈ Univ ∧ 𝑥𝑦) ↔ ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡𝑣𝑡𝑧) → ¬ (𝑣𝑦 → ¬ 𝑣𝑤)) → ¬ ∀𝑖(𝑖𝑧 → (𝑣𝑦 → (𝑖𝑣 → (𝑣𝑓 → ¬ ∀𝑢(𝑢𝑓 → (𝑖𝑢 → ¬ ∀𝑜(𝑜𝑢 → ∀𝑠(𝑠𝑜𝑠𝑤)))))))))))))
1110expandexn 43649 . . 3 (∃𝑦(𝑦 ∈ Univ ∧ 𝑥𝑦) ↔ ¬ ∀𝑦(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡𝑣𝑡𝑧) → ¬ (𝑣𝑦 → ¬ 𝑣𝑤)) → ¬ ∀𝑖(𝑖𝑧 → (𝑣𝑦 → (𝑖𝑣 → (𝑣𝑓 → ¬ ∀𝑢(𝑢𝑓 → (𝑖𝑢 → ¬ ∀𝑜(𝑜𝑢 → ∀𝑠(𝑠𝑜𝑠𝑤)))))))))))))
121, 11bitri 275 . 2 (∃𝑦 ∈ Univ 𝑥𝑦 ↔ ¬ ∀𝑦(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡𝑣𝑡𝑧) → ¬ (𝑣𝑦 → ¬ 𝑣𝑤)) → ¬ ∀𝑖(𝑖𝑧 → (𝑣𝑦 → (𝑖𝑣 → (𝑣𝑓 → ¬ ∀𝑢(𝑢𝑓 → (𝑖𝑢 → ¬ ∀𝑜(𝑜𝑢 → ∀𝑠(𝑠𝑜𝑠𝑤)))))))))))))
1312albii 1814 1 (∀𝑥𝑦 ∈ Univ 𝑥𝑦 ↔ ∀𝑥 ¬ ∀𝑦(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡𝑣𝑡𝑧) → ¬ (𝑣𝑦 → ¬ 𝑣𝑤)) → ¬ ∀𝑖(𝑖𝑧 → (𝑣𝑦 → (𝑖𝑣 → (𝑣𝑓 → ¬ ∀𝑢(𝑢𝑓 → (𝑖𝑢 → ¬ ∀𝑜(𝑜𝑢 → ∀𝑠(𝑠𝑜𝑠𝑤)))))))))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wal 1532  wex 1774  wcel 2099  wral 3056  wrex 3065  Vcvv 3469  wss 3944  𝒫 cpw 4598   cuni 4903  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
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-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-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-tc 9752  df-r1 9779  df-rank 9780  df-card 9954  df-cf 9956  df-acn 9957  df-ac 10131  df-wina 10699  df-ina 10700  df-gru 10806  df-scott 43596  df-coll 43611
This theorem is referenced by:  rr-grothprim  43660
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