Mathbox for Rohan Ridenour < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rr-grothprim Structured version   Visualization version   GIF version

Theorem rr-grothprim 40942
 Description: An equivalent of ax-groth 10234 using only primitives. This uses only 123 symbols, which is significantly less than the previous record of 163 established by grothprim 10245 (which uses some defined symbols, and requires 229 symbols if expanded to primitives). (Contributed by Rohan Ridenour, 13-Aug-2023.)
Assertion
Ref Expression
rr-grothprim ¬ ∀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:   𝑥,𝑦   𝑢,𝑜   𝑧,𝑣,𝑡   𝑤,𝑜,𝑠   𝑦,𝑧,𝑤,𝑣,𝑢,𝑓,𝑖

Proof of Theorem rr-grothprim
StepHypRef Expression
1 gruex 40940 . . . 4 𝑦 ∈ Univ 𝑥𝑦
21ax-gen 1797 . . 3 𝑥𝑦 ∈ Univ 𝑥𝑦
3 rr-grothprimbi 40937 . . 3 (∀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)))))))))))))
42, 3mpbi 233 . 2 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))))))))))))
54spi 2184 1 ¬ ∀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  ∀wal 1536   ∈ wcel 2114  ∃wrex 3131  Univcgru 10201 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-reg 9044  ax-inf2 9092  ax-ac2 9874  ax-groth 10234 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-smo 7970  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-oi 8962  df-har 9009  df-tc 9167  df-r1 9181  df-rank 9182  df-card 9356  df-aleph 9357  df-cf 9358  df-acn 9359  df-ac 9531  df-wina 10095  df-ina 10096  df-tsk 10160  df-gru 10202  df-scott 40878  df-coll 40893 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator