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Theorem rr-grothprimbi 40786
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 40791. (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 3131 . . 3 (∃𝑦 ∈ Univ 𝑥𝑦 ↔ ∃𝑦(𝑦 ∈ Univ ∧ 𝑥𝑦))
2 ancom 463 . . . . 5 ((𝑦 ∈ Univ ∧ 𝑥𝑦) ↔ (𝑥𝑦𝑦 ∈ Univ))
3 biid 263 . . . . . 6 (𝑥𝑦𝑥𝑦)
4 grumnueq 40778 . . . . . . . . 9 Univ = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
54ismnu 40752 . . . . . . . 8 (𝑦 ∈ V → (𝑦 ∈ Univ ↔ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∀𝑓𝑤𝑦 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑦 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))))
65elv 3478 . . . . . . 7 (𝑦 ∈ Univ ↔ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∀𝑓𝑤𝑦 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑦 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
7 ismnuprim 40785 . . . . . . 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 277 . . . . . 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 40779 . . . . 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 277 . . . 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 40780 . . 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 277 . 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 1820 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 208  wa 398  wal 1535  wex 1780  wcel 2114  wral 3125  wrex 3126  Vcvv 3473  wss 3913  𝒫 cpw 4515   cuni 4814  Univcgru 10190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-reg 9034  ax-inf2 9082  ax-ac2 9863
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-iin 4898  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-se 5491  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-isom 6340  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-er 8267  df-map 8386  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-tc 9157  df-r1 9171  df-rank 9172  df-card 9346  df-cf 9348  df-acn 9349  df-ac 9520  df-wina 10084  df-ina 10085  df-gru 10191  df-scott 40727  df-coll 40742
This theorem is referenced by:  rr-grothprim  40791
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