Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  gonanegoal Structured version   Visualization version   GIF version

Theorem gonanegoal 34642
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 (𝑎𝑔𝑏) ≠ ∀𝑔𝑖𝑢

Proof of Theorem gonanegoal
StepHypRef Expression
1 1one2o 8649 . . . 4 1o ≠ 2o
21neii 2941 . . 3 ¬ 1o = 2o
32intnanr 487 . 2 ¬ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩)
4 gonafv 34640 . . . . . 6 ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (𝑎𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩)
54el2v 3481 . . . . 5 (𝑎𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩
6 df-goal 34632 . . . . 5 𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
75, 6eqeq12i 2749 . . . 4 ((𝑎𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ ⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
8 1oex 8480 . . . . 5 1o ∈ V
9 opex 5464 . . . . 5 𝑎, 𝑏⟩ ∈ V
108, 9opth 5476 . . . 4 (⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩))
117, 10bitri 275 . . 3 ((𝑎𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩))
1211necon3abii 2986 . 2 ((𝑎𝑔𝑏) ≠ ∀𝑔𝑖𝑢 ↔ ¬ (1o = 2o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑖, 𝑢⟩))
133, 12mpbir 230 1 (𝑎𝑔𝑏) ≠ ∀𝑔𝑖𝑢
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1540  wne 2939  Vcvv 3473  cop 4634  (class class class)co 7412  1oc1o 8463  2oc2o 8464  𝑔cgna 34624  𝑔cgol 34625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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 7415  df-om 7860  df-1o 8470  df-2o 8471  df-gona 34631  df-goal 34632
This theorem is referenced by:  gonarlem  34684  gonar  34685  goalrlem  34686  goalr  34687  fmlasucdisj  34689
  Copyright terms: Public domain W3C validator