Theorem gonanegoal 35337

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 8683	. . . 4 ⊢ 1o ≠ 2o
2	1	neii 2940	. . 3 ⊢ ¬ 1o = 2o
3	2	intnanr 487	. 2 ⊢ ¬ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩)
4		gonafv 35335	. . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (𝑎⊼𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩)
5	4	el2v 3485	. . . . 5 ⊢ (𝑎⊼𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩
6		df-goal 35327	. . . . 5 ⊢ ∀𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
7	5, 6	eqeq12i 2753	. . . 4 ⊢ ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ ⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
8		1oex 8515	. . . . 5 ⊢ 1o ∈ V
9		opex 5475	. . . . 5 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
10	8, 9	opth 5487	. . . 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 2985	. 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 2938  Vcvv 3478  ⟨cop 4637  (class class class)co 7431  1_oc1o 8498  2_oc2o 8499  ⊼_𝑔cgna 35319  ∀_𝑔cgol 35320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-om 7888  df-1o 8505  df-2o 8506  df-gona 35326  df-goal 35327

This theorem is referenced by:  gonarlem  35379  gonar  35380  goalrlem  35381  goalr  35382  fmlasucdisj  35384