Theorem gonanegoal 35554

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 8577	. . . 4 ⊢ 1o ≠ 2o
2	1	neii 2935	. . 3 ⊢ ¬ 1o = 2o
3	2	intnanr 487	. 2 ⊢ ¬ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩)
4		gonafv 35552	. . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (𝑎⊼𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩)
5	4	el2v 3437	. . . . 5 ⊢ (𝑎⊼𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩
6		df-goal 35544	. . . . 5 ⊢ ∀𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
7	5, 6	eqeq12i 2755	. . . 4 ⊢ ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ ⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
8		1oex 8410	. . . . 5 ⊢ 1o ∈ V
9		opex 5413	. . . . 5 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
10	8, 9	opth 5426	. . . 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 2979	. 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 1542   ≠ wne 2933  Vcvv 3430  ⟨cop 4574  (class class class)co 7362  1_oc1o 8393  2_oc2o 8394  ⊼_𝑔cgna 35536  ∀_𝑔cgol 35537

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fv 6502  df-ov 7365  df-om 7813  df-1o 8400  df-2o 8401  df-gona 35543  df-goal 35544

This theorem is referenced by:  gonarlem  35596  gonar  35597  goalrlem  35598  goalr  35599  fmlasucdisj  35601