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Theorem fmlaomn0 34210
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 6878 . . . . 5 (𝑥 = ∅ → (Fmla‘𝑥) = (Fmla‘∅))
21eleq2d 2818 . . . 4 (𝑥 = ∅ → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘∅)))
32notbid 317 . . 3 (𝑥 = ∅ → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘∅)))
4 fveq2 6878 . . . . 5 (𝑥 = 𝑦 → (Fmla‘𝑥) = (Fmla‘𝑦))
54eleq2d 2818 . . . 4 (𝑥 = 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑦)))
65notbid 317 . . 3 (𝑥 = 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑦)))
7 fveq2 6878 . . . . 5 (𝑥 = suc 𝑦 → (Fmla‘𝑥) = (Fmla‘suc 𝑦))
87eleq2d 2818 . . . 4 (𝑥 = suc 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘suc 𝑦)))
98notbid 317 . . 3 (𝑥 = suc 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘suc 𝑦)))
10 fveq2 6878 . . . . 5 (𝑥 = 𝑁 → (Fmla‘𝑥) = (Fmla‘𝑁))
1110eleq2d 2818 . . . 4 (𝑥 = 𝑁 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑁)))
1211notbid 317 . . 3 (𝑥 = 𝑁 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑁)))
13 0ex 5300 . . . . . . . . . . . 12 ∅ ∈ V
14 opex 5457 . . . . . . . . . . . 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 2993 . . . . . . . . . . 11 (∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ≠ ∅)
18 opnz 5466 . . . . . . . . . . 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 2944 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
22 goel 34167 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
2322eqeq2d 2742 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (∅ = (𝑖𝑔𝑗) ↔ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
2421, 23mtbird 324 . . . . . . 7 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = (𝑖𝑔𝑗))
2524rgen2 3196 . . . . . 6 𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ∅ = (𝑖𝑔𝑗)
26 ralnex2 3132 . . . . . 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 34202 . . . . . 6 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}
3029eleq2i 2824 . . . . 5 (∅ ∈ (Fmla‘∅) ↔ ∅ ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)})
31 eqeq1 2735 . . . . . . 7 (𝑥 = ∅ → (𝑥 = (𝑖𝑔𝑗) ↔ ∅ = (𝑖𝑔𝑗)))
32312rexbidv 3218 . . . . . 6 (𝑥 = ∅ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗)))
3332elrab 3679 . . . . 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 8458 . . . . . . . . . . . . . 14 1o ∈ V
38 opex 5457 . . . . . . . . . . . . . 14 𝑢, 𝑣⟩ ∈ V
3937, 38opnzi 5467 . . . . . . . . . . . . 13 ⟨1o, ⟨𝑢, 𝑣⟩⟩ ≠ ∅
4039nesymi 2997 . . . . . . . . . . . 12 ¬ ∅ = ⟨1o, ⟨𝑢, 𝑣⟩⟩
41 gonafv 34170 . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . 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 3145 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → ∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣))
46 2oex 8459 . . . . . . . . . . . . . . 15 2o ∈ V
47 opex 5457 . . . . . . . . . . . . . . 15 𝑖, 𝑢⟩ ∈ V
4846, 47opnzi 5467 . . . . . . . . . . . . . 14 ⟨2o, ⟨𝑖, 𝑢⟩⟩ ≠ ∅
4948nesymi 2997 . . . . . . . . . . . . 13 ¬ ∅ = ⟨2o, ⟨𝑖, 𝑢⟩⟩
50 df-goal 34162 . . . . . . . . . . . . . 14 𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
5150eqeq2i 2744 . . . . . . . . . . . . 13 (∅ = ∀𝑔𝑖𝑢 ↔ ∅ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
5249, 51mtbir 322 . . . . . . . . . . . 12 ¬ ∅ = ∀𝑔𝑖𝑢
5352a1i 11 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑖 ∈ ω) → ¬ ∅ = ∀𝑔𝑖𝑢)
5453ralrimiva 3145 . . . . . . . . . 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 3145 . . . . . . . 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 3071 . . . . . . . . . . 11 (∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ↔ ¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣))
59 ralnex 3071 . . . . . . . . . . 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 3092 . . . . . . . 8 (∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢) ↔ ∀𝑢 ∈ (Fmla‘𝑦) ¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
64 ralnex 3071 . . . . . . . 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 34206 . . . . . . . 8 (𝑦 ∈ ω → (Fmla‘suc 𝑦) = ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
7069eleq2d 2818 . . . . . . 7 (𝑦 ∈ ω → (∅ ∈ (Fmla‘suc 𝑦) ↔ ∅ ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)})))
71 elun 4144 . . . . . . . 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 2735 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥 = (𝑢𝑔𝑣) ↔ ∅ = (𝑢𝑔𝑣)))
7372rexbidv 3177 . . . . . . . . . . . 12 (𝑥 = ∅ → (∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣)))
74 eqeq1 2735 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥 = ∀𝑔𝑖𝑢 ↔ ∅ = ∀𝑔𝑖𝑢))
7574rexbidv 3177 . . . . . . . . . . . 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 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 3664 . . . . . . . . 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 7871 . 2 (𝑁 ∈ ω → ¬ ∅ ∈ (Fmla‘𝑁))
86 df-nel 3046 . 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 2708  wne 2939  wnel 3045  wral 3060  wrex 3069  {crab 3431  Vcvv 3473  cun 3942  c0 4318  cop 4628  suc csuc 6355  cfv 6532  (class class class)co 7393  ωcom 7838  1oc1o 8441  2oc2o 8442  𝑔cgoe 34153  𝑔cgna 34154  𝑔cgol 34155  Fmlacfmla 34157
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-inf2 9618
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-2o 8449  df-map 8805  df-goel 34160  df-gona 34161  df-goal 34162  df-sat 34163  df-fmla 34165
This theorem is referenced by:  fmlan0  34211  gonan0  34212
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