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Theorem fmlaomn0 33984
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 6842 . . . . 5 (𝑥 = ∅ → (Fmla‘𝑥) = (Fmla‘∅))
21eleq2d 2823 . . . 4 (𝑥 = ∅ → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘∅)))
32notbid 317 . . 3 (𝑥 = ∅ → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘∅)))
4 fveq2 6842 . . . . 5 (𝑥 = 𝑦 → (Fmla‘𝑥) = (Fmla‘𝑦))
54eleq2d 2823 . . . 4 (𝑥 = 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑦)))
65notbid 317 . . 3 (𝑥 = 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑦)))
7 fveq2 6842 . . . . 5 (𝑥 = suc 𝑦 → (Fmla‘𝑥) = (Fmla‘suc 𝑦))
87eleq2d 2823 . . . 4 (𝑥 = suc 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘suc 𝑦)))
98notbid 317 . . 3 (𝑥 = suc 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘suc 𝑦)))
10 fveq2 6842 . . . . 5 (𝑥 = 𝑁 → (Fmla‘𝑥) = (Fmla‘𝑁))
1110eleq2d 2823 . . . 4 (𝑥 = 𝑁 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑁)))
1211notbid 317 . . 3 (𝑥 = 𝑁 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑁)))
13 0ex 5264 . . . . . . . . . . . 12 ∅ ∈ V
14 opex 5421 . . . . . . . . . . . 12 𝑖, 𝑗⟩ ∈ V
1513, 14pm3.2i 471 . . . . . . . . . . 11 (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V)
1615a1i 11 . . . . . . . . . 10 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
17 necom 2997 . . . . . . . . . . 11 (∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ≠ ∅)
18 opnz 5430 . . . . . . . . . . 11 (⟨∅, ⟨𝑖, 𝑗⟩⟩ ≠ ∅ ↔ (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
1917, 18bitri 274 . . . . . . . . . 10 (∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ (∅ ∈ V ∧ ⟨𝑖, 𝑗⟩ ∈ V))
2016, 19sylibr 233 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩)
2120neneqd 2948 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
22 goel 33941 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
2322eqeq2d 2747 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (∅ = (𝑖𝑔𝑗) ↔ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
2421, 23mtbird 324 . . . . . . 7 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = (𝑖𝑔𝑗))
2524rgen2 3194 . . . . . 6 𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ∅ = (𝑖𝑔𝑗)
26 ralnex2 3130 . . . . . 6 (∀𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ∅ = (𝑖𝑔𝑗) ↔ ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗))
2725, 26mpbi 229 . . . . 5 ¬ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗)
2827intnan 487 . . . 4 ¬ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗))
29 fmla0 33976 . . . . . 6 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}
3029eleq2i 2829 . . . . 5 (∅ ∈ (Fmla‘∅) ↔ ∅ ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)})
31 eqeq1 2740 . . . . . . 7 (𝑥 = ∅ → (𝑥 = (𝑖𝑔𝑗) ↔ ∅ = (𝑖𝑔𝑗)))
32312rexbidv 3213 . . . . . 6 (𝑥 = ∅ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗)))
3332elrab 3645 . . . . 5 (∅ ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)} ↔ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗)))
3430, 33bitri 274 . . . 4 (∅ ∈ (Fmla‘∅) ↔ (∅ ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗)))
3528, 34mtbir 322 . . 3 ¬ ∅ ∈ (Fmla‘∅)
36 simpr 485 . . . . . 6 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∅ ∈ (Fmla‘𝑦))
37 1oex 8422 . . . . . . . . . . . . . 14 1o ∈ V
38 opex 5421 . . . . . . . . . . . . . 14 𝑢, 𝑣⟩ ∈ V
3937, 38opnzi 5431 . . . . . . . . . . . . 13 ⟨1o, ⟨𝑢, 𝑣⟩⟩ ≠ ∅
4039nesymi 3001 . . . . . . . . . . . 12 ¬ ∅ = ⟨1o, ⟨𝑢, 𝑣⟩⟩
41 gonafv 33944 . . . . . . . . . . . . . 14 ((𝑢 ∈ (Fmla‘𝑦) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (𝑢𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
4241adantll 712 . . . . . . . . . . . . 13 (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (𝑢𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
4342eqeq2d 2747 . . . . . . . . . . . 12 (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → (∅ = (𝑢𝑔𝑣) ↔ ∅ = ⟨1o, ⟨𝑢, 𝑣⟩⟩))
4440, 43mtbiri 326 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑣 ∈ (Fmla‘𝑦)) → ¬ ∅ = (𝑢𝑔𝑣))
4544ralrimiva 3143 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → ∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣))
46 2oex 8423 . . . . . . . . . . . . . . 15 2o ∈ V
47 opex 5421 . . . . . . . . . . . . . . 15 𝑖, 𝑢⟩ ∈ V
4846, 47opnzi 5431 . . . . . . . . . . . . . 14 ⟨2o, ⟨𝑖, 𝑢⟩⟩ ≠ ∅
4948nesymi 3001 . . . . . . . . . . . . 13 ¬ ∅ = ⟨2o, ⟨𝑖, 𝑢⟩⟩
50 df-goal 33936 . . . . . . . . . . . . . 14 𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
5150eqeq2i 2749 . . . . . . . . . . . . 13 (∅ = ∀𝑔𝑖𝑢 ↔ ∅ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
5249, 51mtbir 322 . . . . . . . . . . . 12 ¬ ∅ = ∀𝑔𝑖𝑢
5352a1i 11 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑖 ∈ ω) → ¬ ∅ = ∀𝑔𝑖𝑢)
5453ralrimiva 3143 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢)
5545, 54jca 512 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → (∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢))
5655ralrimiva 3143 . . . . . . . 8 (𝑦 ∈ ω → ∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢))
5756adantr 481 . . . . . . 7 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢))
58 ralnex 3075 . . . . . . . . . . 11 (∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ↔ ¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣))
59 ralnex 3075 . . . . . . . . . . 11 (∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢 ↔ ¬ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢)
6058, 59anbi12i 627 . . . . . . . . . 10 ((∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
61 ioran 982 . . . . . . . . . 10 (¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
6260, 61bitr4i 277 . . . . . . . . 9 ((∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢) ↔ ¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
6362ralbii 3096 . . . . . . . 8 (∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢) ↔ ∀𝑢 ∈ (Fmla‘𝑦) ¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
64 ralnex 3075 . . . . . . . 8 (∀𝑢 ∈ (Fmla‘𝑦) ¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢) ↔ ¬ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
6563, 64bitri 274 . . . . . . 7 (∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢) ↔ ¬ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
6657, 65sylib 217 . . . . . 6 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
67 ioran 982 . . . . . 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 33980 . . . . . . . 8 (𝑦 ∈ ω → (Fmla‘suc 𝑦) = ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
7069eleq2d 2823 . . . . . . 7 (𝑦 ∈ ω → (∅ ∈ (Fmla‘suc 𝑦) ↔ ∅ ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)})))
71 elun 4108 . . . . . . . 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 2740 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥 = (𝑢𝑔𝑣) ↔ ∅ = (𝑢𝑔𝑣)))
7372rexbidv 3175 . . . . . . . . . . . 12 (𝑥 = ∅ → (∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣)))
74 eqeq1 2740 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥 = ∀𝑔𝑖𝑢 ↔ ∅ = ∀𝑔𝑖𝑢))
7574rexbidv 3175 . . . . . . . . . . . 12 (𝑥 = ∅ → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
7673, 75orbi12d 917 . . . . . . . . . . 11 (𝑥 = ∅ → ((∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢)))
7776rexbidv 3175 . . . . . . . . . 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 3630 . . . . . . . . 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 911 . . . . . . . 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 274 . . . . . . 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 286 . . . . . 6 (𝑦 ∈ ω → (∅ ∈ (Fmla‘suc 𝑦) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))))
8281adantr 481 . . . . 5 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → (∅ ∈ (Fmla‘suc 𝑦) ↔ (∅ ∈ (Fmla‘𝑦) ∨ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))))
8368, 82mtbird 324 . . . 4 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ (Fmla‘𝑦)) → ¬ ∅ ∈ (Fmla‘suc 𝑦))
8483ex 413 . . 3 (𝑦 ∈ ω → (¬ ∅ ∈ (Fmla‘𝑦) → ¬ ∅ ∈ (Fmla‘suc 𝑦)))
853, 6, 9, 12, 35, 84finds 7835 . 2 (𝑁 ∈ ω → ¬ ∅ ∈ (Fmla‘𝑁))
86 df-nel 3050 . 2 (∅ ∉ (Fmla‘𝑁) ↔ ¬ ∅ ∈ (Fmla‘𝑁))
8785, 86sylibr 233 1 (𝑁 ∈ ω → ∅ ∉ (Fmla‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  {cab 2713  wne 2943  wnel 3049  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cun 3908  c0 4282  cop 4592  suc csuc 6319  cfv 6496  (class class class)co 7357  ωcom 7802  1oc1o 8405  2oc2o 8406  𝑔cgoe 33927  𝑔cgna 33928  𝑔cgol 33929  Fmlacfmla 33931
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-map 8767  df-goel 33934  df-gona 33935  df-goal 33936  df-sat 33937  df-fmla 33939
This theorem is referenced by:  fmlan0  33985  gonan0  33986
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