Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fmlaomn0 Structured version   Visualization version   GIF version

Theorem fmlaomn0 35375
Description: The empty set is not a Godel formula of any height. (Contributed by AV, 21-Oct-2023.)
Assertion
Ref Expression
fmlaomn0 (𝑁 ∈ ω → ∅ ∉ (Fmla‘𝑁))

Proof of Theorem fmlaomn0
Dummy variables 𝑥 𝑖 𝑗 𝑢 𝑣 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . . 5 (𝑥 = ∅ → (Fmla‘𝑥) = (Fmla‘∅))
21eleq2d 2825 . . . 4 (𝑥 = ∅ → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘∅)))
32notbid 318 . . 3 (𝑥 = ∅ → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘∅)))
4 fveq2 6907 . . . . 5 (𝑥 = 𝑦 → (Fmla‘𝑥) = (Fmla‘𝑦))
54eleq2d 2825 . . . 4 (𝑥 = 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑦)))
65notbid 318 . . 3 (𝑥 = 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑦)))
7 fveq2 6907 . . . . 5 (𝑥 = suc 𝑦 → (Fmla‘𝑥) = (Fmla‘suc 𝑦))
87eleq2d 2825 . . . 4 (𝑥 = suc 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘suc 𝑦)))
98notbid 318 . . 3 (𝑥 = suc 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘suc 𝑦)))
10 fveq2 6907 . . . . 5 (𝑥 = 𝑁 → (Fmla‘𝑥) = (Fmla‘𝑁))
1110eleq2d 2825 . . . 4 (𝑥 = 𝑁 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑁)))
1211notbid 318 . . 3 (𝑥 = 𝑁 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑁)))
13 0ex 5313 . . . . . . . . . . . 12 ∅ ∈ V
14 opex 5475 . . . . . . . . . . . 12 𝑖, 𝑗⟩ ∈ V
1513, 14pm3.2i 470 . . . . . . . . . . 11 (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V)
1615a1i 11 . . . . . . . . . 10 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
17 necom 2992 . . . . . . . . . . 11 (∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ≠ ∅)
18 opnz 5484 . . . . . . . . . . 11 (⟨∅, ⟨𝑖, 𝑗⟩⟩ ≠ ∅ ↔ (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
1917, 18bitri 275 . . . . . . . . . 10 (∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
2016, 19sylibr 234 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩)
2120neneqd 2943 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
22 goel 35332 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
2322eqeq2d 2746 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (∅ = (𝑖𝑔𝑗) ↔ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
2421, 23mtbird 325 . . . . . . 7 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = (𝑖𝑔𝑗))
2524rgen2 3197 . . . . . 6 𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ∅ = (𝑖𝑔𝑗)
26 ralnex2 3131 . . . . . 6 (∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ∅ = (𝑖𝑔𝑗) ↔ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗))
2725, 26mpbi 230 . . . . 5 ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗)
2827intnan 486 . . . 4 ¬ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗))
29 fmla0 35367 . . . . . 6 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}
3029eleq2i 2831 . . . . 5 (∅ ∈ (Fmla‘∅) ↔ ∅ ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)})
31 eqeq1 2739 . . . . . . 7 (𝑥 = ∅ → (𝑥 = (𝑖𝑔𝑗) ↔ ∅ = (𝑖𝑔𝑗)))
32312rexbidv 3220 . . . . . 6 (𝑥 = ∅ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗)))
3332elrab 3695 . . . . 5 (∅ ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)} ↔ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗)))
3430, 33bitri 275 . . . 4 (∅ ∈ (Fmla‘∅) ↔ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗)))
3528, 34mtbir 323 . . 3 ¬ ∅ ∈ (Fmla‘∅)
36 simpr 484 . . . . . 6 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∅ ∈ (Fmla‘𝑦))
37 1oex 8515 . . . . . . . . . . . . . 14 1o ∈ V
38 opex 5475 . . . . . . . . . . . . . 14 𝑢, 𝑣⟩ ∈ V
3937, 38opnzi 5485 . . . . . . . . . . . . 13 ⟨1o, ⟨𝑢, 𝑣⟩⟩ ≠ ∅
4039nesymi 2996 . . . . . . . . . . . 12 ¬ ∅ = ⟨1o, ⟨𝑢, 𝑣⟩⟩
41 gonafv 35335 . . . . . . . . . . . . . 14 ((𝑢 ∈ (Fmla‘𝑦) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (𝑢𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
4241adantll 714 . . . . . . . . . . . . 13 (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (𝑢𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
4342eqeq2d 2746 . . . . . . . . . . . 12 (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (∅ = (𝑢𝑔𝑣) ↔ ∅ = ⟨1o, ⟨𝑢, 𝑣⟩⟩))
4440, 43mtbiri 327 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → ¬ ∅ = (𝑢𝑔𝑣))
4544ralrimiva 3144 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → ∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣))
46 2oex 8516 . . . . . . . . . . . . . . 15 2o ∈ V
47 opex 5475 . . . . . . . . . . . . . . 15 𝑖, 𝑢⟩ ∈ V
4846, 47opnzi 5485 . . . . . . . . . . . . . 14 ⟨2o, ⟨𝑖, 𝑢⟩⟩ ≠ ∅
4948nesymi 2996 . . . . . . . . . . . . 13 ¬ ∅ = ⟨2o, ⟨𝑖, 𝑢⟩⟩
50 df-goal 35327 . . . . . . . . . . . . . 14 𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
5150eqeq2i 2748 . . . . . . . . . . . . 13 (∅ = ∀𝑔𝑖𝑢 ↔ ∅ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
5249, 51mtbir 323 . . . . . . . . . . . 12 ¬ ∅ = ∀𝑔𝑖𝑢
5352a1i 11 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑖 ∈ ω) → ¬ ∅ = ∀𝑔𝑖𝑢)
5453ralrimiva 3144 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢)
5545, 54jca 511 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → (∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢))
5655ralrimiva 3144 . . . . . . . 8 (𝑦 ∈ ω → ∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢))
5756adantr 480 . . . . . . 7 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢))
58 ralnex 3070 . . . . . . . . . . 11 (∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ↔ ¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣))
59 ralnex 3070 . . . . . . . . . . 11 (∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢 ↔ ¬ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢)
6058, 59anbi12i 628 . . . . . . . . . 10 ((∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
61 ioran 985 . . . . . . . . . 10 (¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
6260, 61bitr4i 278 . . . . . . . . 9 ((∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢) ↔ ¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
6362ralbii 3091 . . . . . . . 8 (∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢) ↔ ∀𝑢 ∈ (Fmla‘𝑦) ¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
64 ralnex 3070 . . . . . . . 8 (∀𝑢 ∈ (Fmla‘𝑦) ¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢) ↔ ¬ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
6563, 64bitri 275 . . . . . . 7 (∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢) ↔ ¬ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
6657, 65sylib 218 . . . . . 6 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
67 ioran 985 . . . . . 6 (¬ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢)) ↔ (¬ ∅ ∈ (Fmla‘𝑦) ∧ ¬ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢)))
6836, 66, 67sylanbrc 583 . . . . 5 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢)))
69 fmlasuc 35371 . . . . . . . 8 (𝑦 ∈ ω → (Fmla‘suc 𝑦) = ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
7069eleq2d 2825 . . . . . . 7 (𝑦 ∈ ω → (∅ ∈ (Fmla‘suc 𝑦) ↔ ∅ ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)})))
71 elun 4163 . . . . . . . 8 (∅ ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∅ ∈ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
72 eqeq1 2739 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥 = (𝑢𝑔𝑣) ↔ ∅ = (𝑢𝑔𝑣)))
7372rexbidv 3177 . . . . . . . . . . . 12 (𝑥 = ∅ → (∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣)))
74 eqeq1 2739 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥 = ∀𝑔𝑖𝑢 ↔ ∅ = ∀𝑔𝑖𝑢))
7574rexbidv 3177 . . . . . . . . . . . 12 (𝑥 = ∅ → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
7673, 75orbi12d 918 . . . . . . . . . . 11 (𝑥 = ∅ → ((∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢)))
7776rexbidv 3177 . . . . . . . . . 10 (𝑥 = ∅ → (∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢)))
7813, 77elab 3681 . . . . . . . . 9 (∅ ∈ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)} ↔ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
7978orbi2i 912 . . . . . . . 8 ((∅ ∈ (Fmla‘𝑦) ∨ ∅ ∈ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢)))
8071, 79bitri 275 . . . . . . 7 (∅ ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢)))
8170, 80bitrdi 287 . . . . . 6 (𝑦 ∈ ω → (∅ ∈ (Fmla‘suc 𝑦) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))))
8281adantr 480 . . . . 5 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → (∅ ∈ (Fmla‘suc 𝑦) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))))
8368, 82mtbird 325 . . . 4 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∅ ∈ (Fmla‘suc 𝑦))
8483ex 412 . . 3 (𝑦 ∈ ω → (¬ ∅ ∈ (Fmla‘𝑦) → ¬ ∅ ∈ (Fmla‘suc 𝑦)))
853, 6, 9, 12, 35, 84finds 7919 . 2 (𝑁 ∈ ω → ¬ ∅ ∈ (Fmla‘𝑁))
86 df-nel 3045 . 2 (∅ ∉ (Fmla‘𝑁) ↔ ¬ ∅ ∈ (Fmla‘𝑁))
8785, 86sylibr 234 1 (𝑁 ∈ ω → ∅ ∉ (Fmla‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  {cab 2712  wne 2938  wnel 3044  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cun 3961  c0 4339  cop 4637  suc csuc 6388  cfv 6563  (class class class)co 7431  ωcom 7887  1oc1o 8498  2oc2o 8499  𝑔cgoe 35318  𝑔cgna 35319  𝑔cgol 35320  Fmlacfmla 35322
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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
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-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  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-iun 4998  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-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-map 8867  df-goel 35325  df-gona 35326  df-goal 35327  df-sat 35328  df-fmla 35330
This theorem is referenced by:  fmlan0  35376  gonan0  35377
  Copyright terms: Public domain W3C validator