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	⊢ (𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢

Proof of Theorem gonanegoal

Step	Hyp	Ref	Expression
1		1one2o 8684	. . . 4 ⊢ 1o ≠ 2o
2	1	neii 2942	. . 3 ⊢ ¬ 1o = 2o
3	2	intnanr 487	. 2 ⊢ ¬ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩)
4		gonafv 35355	. . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (𝑎⊼𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩)
5	4	el2v 3487	. . . . 5 ⊢ (𝑎⊼𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩
6		df-goal 35347	. . . . 5 ⊢ ∀𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
7	5, 6	eqeq12i 2755	. . . 4 ⊢ ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ ⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
8		1oex 8516	. . . . 5 ⊢ 1o ∈ V
9		opex 5469	. . . . 5 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
10	8, 9	opth 5481	. . . 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 2987	. 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 1540   ≠ wne 2940  Vcvv 3480  ⟨cop 4632  (class class class)co 7431  1_oc1o 8499  2_oc2o 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