Theorem rr-grothprimbi 45278

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 45283. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Assertion

Ref	Expression
rr-grothprimbi	⊢ (∀𝑥∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 ↔ ∀𝑥 ¬ ∀𝑦(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑦 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))))))

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.

Step	Hyp	Ref	Expression
1		df-rex 3088	. . 3 ⊢ (∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 ↔ ∃𝑦(𝑦 ∈ Univ ∧ 𝑥 ∈ 𝑦))
2		ancom 466	. . . . 5 ⊢ ((𝑦 ∈ Univ ∧ 𝑥 ∈ 𝑦) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ Univ))
3		biid 264	. . . . . 6 ⊢ (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)
4		grumnueq 45270	. . . . . . . . 9 ⊢ Univ = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
5	4	ismnu 45244	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑦 ∈ Univ ↔ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∀𝑓∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑦 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))))
6	5	elv 3456	. . . . . . 7 ⊢ (𝑦 ∈ Univ ↔ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∀𝑓∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑦 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
7		ismnuprim 45277	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∀𝑓∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑦 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑦 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))))))))))
8	6, 7	bitri 278	. . . . . 6 ⊢ (𝑦 ∈ Univ ↔ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑦 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))))))))))
9	3, 8	expandan 45271	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ Univ) ↔ ¬ (𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑦 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))))))
10	2, 9	bitri 278	. . . 4 ⊢ ((𝑦 ∈ Univ ∧ 𝑥 ∈ 𝑦) ↔ ¬ (𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑦 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))))))
11	10	expandexn 45272	. . 3 ⊢ (∃𝑦(𝑦 ∈ Univ ∧ 𝑥 ∈ 𝑦) ↔ ¬ ∀𝑦(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑦 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))))))
12	1, 11	bitri 278	. 2 ⊢ (∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 ↔ ¬ ∀𝑦(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑦 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))))))
13	12	albii 1852	1 ⊢ (∀𝑥∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 ↔ ∀𝑥 ¬ ∀𝑦(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑦 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  Univcgru 10875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-reg 9586  ax-inf2 9642  ax-ac2 10541

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-tc 9736  df-r1 9768  df-rank 9769  df-scott 9929  df-card 10020  df-cf 10022  df-acn 10023  df-ac 10195  df-wina 10769  df-ina 10770  df-gru 10876  df-coll 45234

This theorem is used by:  rr-grothprim  45283