Theorem gonanegoal 35320

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	⊢ (𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢

Proof of Theorem gonanegoal

Step	Hyp	Ref	Expression
1		1one2o 8702	. . . 4 ⊢ 1o ≠ 2o
2	1	neii 2948	. . 3 ⊢ ¬ 1o = 2o
3	2	intnanr 487	. 2 ⊢ ¬ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩)
4		gonafv 35318	. . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (𝑎⊼𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩)
5	4	el2v 3495	. . . . 5 ⊢ (𝑎⊼𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩
6		df-goal 35310	. . . . 5 ⊢ ∀𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
7	5, 6	eqeq12i 2758	. . . 4 ⊢ ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ ⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
8		1oex 8532	. . . . 5 ⊢ 1o ∈ V
9		opex 5484	. . . . 5 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
10	8, 9	opth 5496	. . . 4 ⊢ (⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩))
11	7, 10	bitri 275	. . 3 ⊢ ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩))
12	11	necon3abii 2993	. 2 ⊢ ((𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢 ↔ ¬ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩))
13	3, 12	mpbir 231	1 ⊢ (𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1537   ≠ wne 2946  Vcvv 3488  ⟨cop 4654  (class class class)co 7448  1_oc1o 8515  2_oc2o 8516  ⊼_𝑔cgna 35302  ∀_𝑔cgol 35303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-om 7904  df-1o 8522  df-2o 8523  df-gona 35309  df-goal 35310

This theorem is referenced by:  gonarlem  35362  gonar  35363  goalrlem  35364  goalr  35365  fmlasucdisj  35367