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Theorem fmlaomn0 35395
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 6906 . . . . 5 (𝑥 = ∅ → (Fmla‘𝑥) = (Fmla‘∅))
21eleq2d 2827 . . . 4 (𝑥 = ∅ → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘∅)))
32notbid 318 . . 3 (𝑥 = ∅ → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘∅)))
4 fveq2 6906 . . . . 5 (𝑥 = 𝑦 → (Fmla‘𝑥) = (Fmla‘𝑦))
54eleq2d 2827 . . . 4 (𝑥 = 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑦)))
65notbid 318 . . 3 (𝑥 = 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑦)))
7 fveq2 6906 . . . . 5 (𝑥 = suc 𝑦 → (Fmla‘𝑥) = (Fmla‘suc 𝑦))
87eleq2d 2827 . . . 4 (𝑥 = suc 𝑦 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘suc 𝑦)))
98notbid 318 . . 3 (𝑥 = suc 𝑦 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘suc 𝑦)))
10 fveq2 6906 . . . . 5 (𝑥 = 𝑁 → (Fmla‘𝑥) = (Fmla‘𝑁))
1110eleq2d 2827 . . . 4 (𝑥 = 𝑁 → (∅ ∈ (Fmla‘𝑥) ↔ ∅ ∈ (Fmla‘𝑁)))
1211notbid 318 . . 3 (𝑥 = 𝑁 → (¬ ∅ ∈ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑁)))
13 0ex 5307 . . . . . . . . . . . 12 ∅ ∈ V
14 opex 5469 . . . . . . . . . . . 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 2994 . . . . . . . . . . 11 (∅ ≠ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ≠ ∅)
18 opnz 5478 . . . . . . . . . . 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 2945 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
22 goel 35352 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
2322eqeq2d 2748 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (∅ = (𝑖𝑔𝑗) ↔ ∅ = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
2421, 23mtbird 325 . . . . . . 7 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ¬ ∅ = (𝑖𝑔𝑗))
2524rgen2 3199 . . . . . 6 𝑖 ∈ ω ∀𝑗 ∈ ω ¬ ∅ = (𝑖𝑔𝑗)
26 ralnex2 3133 . . . . . 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 35387 . . . . . 6 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}
3029eleq2i 2833 . . . . 5 (∅ ∈ (Fmla‘∅) ↔ ∅ ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)})
31 eqeq1 2741 . . . . . . 7 (𝑥 = ∅ → (𝑥 = (𝑖𝑔𝑗) ↔ ∅ = (𝑖𝑔𝑗)))
32312rexbidv 3222 . . . . . 6 (𝑥 = ∅ → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∅ = (𝑖𝑔𝑗)))
3332elrab 3692 . . . . 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 8516 . . . . . . . . . . . . . 14 1o ∈ V
38 opex 5469 . . . . . . . . . . . . . 14 𝑢, 𝑣⟩ ∈ V
3937, 38opnzi 5479 . . . . . . . . . . . . 13 ⟨1o, ⟨𝑢, 𝑣⟩⟩ ≠ ∅
4039nesymi 2998 . . . . . . . . . . . 12 ¬ ∅ = ⟨1o, ⟨𝑢, 𝑣⟩⟩
41 gonafv 35355 . . . . . . . . . . . . . 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 2748 . . . . . . . . . . . 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 3146 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) → ∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣))
46 2oex 8517 . . . . . . . . . . . . . . 15 2o ∈ V
47 opex 5469 . . . . . . . . . . . . . . 15 𝑖, 𝑢⟩ ∈ V
4846, 47opnzi 5479 . . . . . . . . . . . . . 14 ⟨2o, ⟨𝑖, 𝑢⟩⟩ ≠ ∅
4948nesymi 2998 . . . . . . . . . . . . 13 ¬ ∅ = ⟨2o, ⟨𝑖, 𝑢⟩⟩
50 df-goal 35347 . . . . . . . . . . . . . 14 𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
5150eqeq2i 2750 . . . . . . . . . . . . 13 (∅ = ∀𝑔𝑖𝑢 ↔ ∅ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
5249, 51mtbir 323 . . . . . . . . . . . 12 ¬ ∅ = ∀𝑔𝑖𝑢
5352a1i 11 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑦)) ∧ 𝑖 ∈ ω) → ¬ ∅ = ∀𝑔𝑖𝑢)
5453ralrimiva 3146 . . . . . . . . . 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 3146 . . . . . . . 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 3072 . . . . . . . . . . 11 (∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ↔ ¬ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣))
59 ralnex 3072 . . . . . . . . . . 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 986 . . . . . . . . . 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 3093 . . . . . . . 8 (∀𝑢 ∈ (Fmla‘𝑦)(∀𝑣 ∈ (Fmla‘𝑦) ¬ ∅ = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ ∅ = ∀𝑔𝑖𝑢) ↔ ∀𝑢 ∈ (Fmla‘𝑦) ¬ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
64 ralnex 3072 . . . . . . . 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 986 . . . . . 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 35391 . . . . . . . 8 (𝑦 ∈ ω → (Fmla‘suc 𝑦) = ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
7069eleq2d 2827 . . . . . . 7 (𝑦 ∈ ω → (∅ ∈ (Fmla‘suc 𝑦) ↔ ∅ ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑦)(∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)})))
71 elun 4153 . . . . . . . 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 2741 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥 = (𝑢𝑔𝑣) ↔ ∅ = (𝑢𝑔𝑣)))
7372rexbidv 3179 . . . . . . . . . . . 12 (𝑥 = ∅ → (∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣)))
74 eqeq1 2741 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥 = ∀𝑔𝑖𝑢 ↔ ∅ = ∀𝑔𝑖𝑢))
7574rexbidv 3179 . . . . . . . . . . . 12 (𝑥 = ∅ → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢))
7673, 75orbi12d 919 . . . . . . . . . . 11 (𝑥 = ∅ → ((∃𝑣 ∈ (Fmla‘𝑦)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ (∃𝑣 ∈ (Fmla‘𝑦)∅ = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω ∅ = ∀𝑔𝑖𝑢)))
7776rexbidv 3179 . . . . . . . . . 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 3679 . . . . . . . . 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 913 . . . . . . . 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 7918 . 2 (𝑁 ∈ ω → ¬ ∅ ∈ (Fmla‘𝑁))
86 df-nel 3047 . 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 848   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wnel 3046  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cun 3949  c0 4333  cop 4632  suc csuc 6386  cfv 6561  (class class class)co 7431  ωcom 7887  1oc1o 8499  2oc2o 8500  𝑔cgoe 35338  𝑔cgna 35339  𝑔cgol 35340  Fmlacfmla 35342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-map 8868  df-goel 35345  df-gona 35346  df-goal 35347  df-sat 35348  df-fmla 35350
This theorem is referenced by:  fmlan0  35396  gonan0  35397
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