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Theorem gonanegoal 35357
Description: The Godel-set for the Sheffer stroke NAND is not equal to the Godel-set of universal quantification. (Contributed by AV, 21-Oct-2023.)
Assertion
Ref Expression
gonanegoal (𝑎𝑔𝑏) ≠ ∀𝑔𝑖𝑢

Proof of Theorem gonanegoal
StepHypRef Expression
1 1one2o 8684 . . . 4 1o ≠ 2o
21neii 2942 . . 3 ¬ 1o = 2o
32intnanr 487 . 2 ¬ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩)
4 gonafv 35355 . . . . . 6 ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (𝑎𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩)
54el2v 3487 . . . . 5 (𝑎𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩
6 df-goal 35347 . . . . 5 𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
75, 6eqeq12i 2755 . . . 4 ((𝑎𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ ⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
8 1oex 8516 . . . . 5 1o ∈ V
9 opex 5469 . . . . 5 𝑎, 𝑏⟩ ∈ V
108, 9opth 5481 . . . 4 (⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩))
117, 10bitri 275 . . 3 ((𝑎𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩))
1211necon3abii 2987 . 2 ((𝑎𝑔𝑏) ≠ ∀𝑔𝑖𝑢 ↔ ¬ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩))
133, 12mpbir 231 1 (𝑎𝑔𝑏) ≠ ∀𝑔𝑖𝑢
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1540  wne 2940  Vcvv 3480  cop 4632  (class class class)co 7431  1oc1o 8499  2oc2o 8500  𝑔cgna 35339  𝑔cgol 35340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-om 7888  df-1o 8506  df-2o 8507  df-gona 35346  df-goal 35347
This theorem is referenced by:  gonarlem  35399  gonar  35400  goalrlem  35401  goalr  35402  fmlasucdisj  35404
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