Theorem gonanegoal 34412

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 8647	. . . 4 ⊢ 1o ≠ 2o
2	1	neii 2942	. . 3 ⊢ ¬ 1o = 2o
3	2	intnanr 488	. 2 ⊢ ¬ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩)
4		gonafv 34410	. . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (𝑎⊼𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩)
5	4	el2v 3482	. . . . 5 ⊢ (𝑎⊼𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩
6		df-goal 34402	. . . . 5 ⊢ ∀𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
7	5, 6	eqeq12i 2750	. . . 4 ⊢ ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ ⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
8		1oex 8478	. . . . 5 ⊢ 1o ∈ V
9		opex 5464	. . . . 5 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
10	8, 9	opth 5476	. . . 4 ⊢ (⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩))
11	7, 10	bitri 274	. . 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 230	1 ⊢ (𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1541   ≠ wne 2940  Vcvv 3474  ⟨cop 4634  (class class class)co 7411  1_oc1o 8461  2_oc2o 8462  ⊼_𝑔cgna 34394  ∀_𝑔cgol 34395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-om 7858  df-1o 8468  df-2o 8469  df-gona 34401  df-goal 34402

This theorem is referenced by:  gonarlem  34454  gonar  34455  goalrlem  34456  goalr  34457  fmlasucdisj  34459