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Theorem gonanegoal 32603
 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 8272 . . . 4 1o ≠ 2o
21neii 3021 . . 3 ¬ 1o = 2o
32intnanr 490 . 2 ¬ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩)
4 gonafv 32601 . . . . . 6 ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (𝑎𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩)
54el2v 3504 . . . . 5 (𝑎𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩
6 df-goal 32593 . . . . 5 𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
75, 6eqeq12i 2839 . . . 4 ((𝑎𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ ⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
8 1oex 8113 . . . . 5 1o ∈ V
9 opex 5359 . . . . 5 𝑎, 𝑏⟩ ∈ V
108, 9opth 5371 . . . 4 (⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩))
117, 10bitri 277 . . 3 ((𝑎𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩))
1211necon3abii 3065 . 2 ((𝑎𝑔𝑏) ≠ ∀𝑔𝑖𝑢 ↔ ¬ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩))
133, 12mpbir 233 1 (𝑎𝑔𝑏) ≠ ∀𝑔𝑖𝑢
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1536   ≠ wne 3019  Vcvv 3497  ⟨cop 4576  (class class class)co 7159  1oc1o 8098  2oc2o 8099  ⊼𝑔cgna 32585  ∀𝑔cgol 32586 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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333  ax-un 7464 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fv 6366  df-ov 7162  df-om 7584  df-1o 8105  df-2o 8106  df-gona 32592  df-goal 32593 This theorem is referenced by:  gonarlem  32645  gonar  32646  goalrlem  32647  goalr  32648  fmlasucdisj  32650
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