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Theorem goalr 32529
Description: If the "Godel-set of universal quantification" applied to a class is a Godel formula, the class is also a Godel formula. Remark: The reverse is not valid for 𝐴 being of the same height as the "Godel-set of universal quantification". (Contributed by AV, 22-Oct-2023.)
Assertion
Ref Expression
goalr ((𝑁 ∈ ω ∧ ∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → 𝑎 ∈ (Fmla‘𝑁))
Distinct variable groups:   𝑖,𝑁   𝑖,𝑎
Allowed substitution hint:   𝑁(𝑎)

Proof of Theorem goalr
Dummy variables 𝑗 𝑥 𝑘 𝑢 𝑣 𝑛 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 goaln0 32525 . . 3 (∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
21adantl 482 . 2 ((𝑁 ∈ ω ∧ ∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → 𝑁 ≠ ∅)
3 nnsuc 7588 . . . 4 ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∃𝑛 ∈ ω 𝑁 = suc 𝑛)
4 suceq 6253 . . . . . . . . . . 11 (𝑥 = ∅ → suc 𝑥 = suc ∅)
54fveq2d 6670 . . . . . . . . . 10 (𝑥 = ∅ → (Fmla‘suc 𝑥) = (Fmla‘suc ∅))
65eleq2d 2902 . . . . . . . . 9 (𝑥 = ∅ → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀𝑔𝑖𝑎 ∈ (Fmla‘suc ∅)))
75eleq2d 2902 . . . . . . . . 9 (𝑥 = ∅ → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc ∅)))
86, 7imbi12d 346 . . . . . . . 8 (𝑥 = ∅ → ((∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) → 𝑎 ∈ (Fmla‘suc ∅))))
9 suceq 6253 . . . . . . . . . . 11 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
109fveq2d 6670 . . . . . . . . . 10 (𝑥 = 𝑦 → (Fmla‘suc 𝑥) = (Fmla‘suc 𝑦))
1110eleq2d 2902 . . . . . . . . 9 (𝑥 = 𝑦 → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑦)))
1210eleq2d 2902 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc 𝑦)))
1311, 12imbi12d 346 . . . . . . . 8 (𝑥 = 𝑦 → ((∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑦) → 𝑎 ∈ (Fmla‘suc 𝑦))))
14 suceq 6253 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1514fveq2d 6670 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (Fmla‘suc 𝑥) = (Fmla‘suc suc 𝑦))
1615eleq2d 2902 . . . . . . . . 9 (𝑥 = suc 𝑦 → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑦)))
1715eleq2d 2902 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc suc 𝑦)))
1816, 17imbi12d 346 . . . . . . . 8 (𝑥 = suc 𝑦 → ((∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑦) → 𝑎 ∈ (Fmla‘suc suc 𝑦))))
19 suceq 6253 . . . . . . . . . . 11 (𝑥 = 𝑛 → suc 𝑥 = suc 𝑛)
2019fveq2d 6670 . . . . . . . . . 10 (𝑥 = 𝑛 → (Fmla‘suc 𝑥) = (Fmla‘suc 𝑛))
2120eleq2d 2902 . . . . . . . . 9 (𝑥 = 𝑛 → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) ↔ ∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛)))
2220eleq2d 2902 . . . . . . . . 9 (𝑥 = 𝑛 → (𝑎 ∈ (Fmla‘suc 𝑥) ↔ 𝑎 ∈ (Fmla‘suc 𝑛)))
2321, 22imbi12d 346 . . . . . . . 8 (𝑥 = 𝑛 → ((∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑥) → 𝑎 ∈ (Fmla‘suc 𝑥)) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛))))
24 peano1 7592 . . . . . . . . . 10 ∅ ∈ ω
25 df-goal 32474 . . . . . . . . . . 11 𝑔𝑖𝑎 = ⟨2o, ⟨𝑖, 𝑎⟩⟩
26 opex 5352 . . . . . . . . . . 11 ⟨2o, ⟨𝑖, 𝑎⟩⟩ ∈ V
2725, 26eqeltri 2913 . . . . . . . . . 10 𝑔𝑖𝑎 ∈ V
28 isfmlasuc 32520 . . . . . . . . . 10 ((∅ ∈ ω ∧ ∀𝑔𝑖𝑎 ∈ V) → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘∅) ∨ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀𝑔𝑖𝑎 = (𝑢𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢))))
2924, 27, 28mp2an 688 . . . . . . . . 9 (∀𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘∅) ∨ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀𝑔𝑖𝑎 = (𝑢𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢)))
30 eqeq1 2828 . . . . . . . . . . . . 13 (𝑥 = ∀𝑔𝑖𝑎 → (𝑥 = (𝑘𝑔𝑗) ↔ ∀𝑔𝑖𝑎 = (𝑘𝑔𝑗)))
31302rexbidv 3304 . . . . . . . . . . . 12 (𝑥 = ∀𝑔𝑖𝑎 → (∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘𝑔𝑗) ↔ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ∀𝑔𝑖𝑎 = (𝑘𝑔𝑗)))
32 fmla0 32514 . . . . . . . . . . . 12 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘𝑔𝑗)}
3331, 32elrab2 3686 . . . . . . . . . . 11 (∀𝑔𝑖𝑎 ∈ (Fmla‘∅) ↔ (∀𝑔𝑖𝑎 ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω ∀𝑔𝑖𝑎 = (𝑘𝑔𝑗)))
3425a1i 11 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → ∀𝑔𝑖𝑎 = ⟨2o, ⟨𝑖, 𝑎⟩⟩)
35 goel 32479 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘𝑔𝑗) = ⟨∅, ⟨𝑘, 𝑗⟩⟩)
3634, 35eqeq12d 2840 . . . . . . . . . . . . . 14 ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (∀𝑔𝑖𝑎 = (𝑘𝑔𝑗) ↔ ⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩))
37 2oex 8106 . . . . . . . . . . . . . . . 16 2o ∈ V
38 opex 5352 . . . . . . . . . . . . . . . 16 𝑖, 𝑎⟩ ∈ V
3937, 38opth 5364 . . . . . . . . . . . . . . 15 (⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩ ↔ (2o = ∅ ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑗⟩))
40 2on0 8107 . . . . . . . . . . . . . . . . 17 2o ≠ ∅
41 eqneqall 3031 . . . . . . . . . . . . . . . . 17 (2o = ∅ → (2o ≠ ∅ → 𝑎 ∈ (Fmla‘suc ∅)))
4240, 41mpi 20 . . . . . . . . . . . . . . . 16 (2o = ∅ → 𝑎 ∈ (Fmla‘suc ∅))
4342adantr 481 . . . . . . . . . . . . . . 15 ((2o = ∅ ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑗⟩) → 𝑎 ∈ (Fmla‘suc ∅))
4439, 43sylbi 218 . . . . . . . . . . . . . 14 (⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨∅, ⟨𝑘, 𝑗⟩⟩ → 𝑎 ∈ (Fmla‘suc ∅))
4536, 44syl6bi 254 . . . . . . . . . . . . 13 ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (∀𝑔𝑖𝑎 = (𝑘𝑔𝑗) → 𝑎 ∈ (Fmla‘suc ∅)))
4645rexlimdva 3288 . . . . . . . . . . . 12 (𝑘 ∈ ω → (∃𝑗 ∈ ω ∀𝑔𝑖𝑎 = (𝑘𝑔𝑗) → 𝑎 ∈ (Fmla‘suc ∅)))
4746rexlimiv 3284 . . . . . . . . . . 11 (∃𝑘 ∈ ω ∃𝑗 ∈ ω ∀𝑔𝑖𝑎 = (𝑘𝑔𝑗) → 𝑎 ∈ (Fmla‘suc ∅))
4833, 47simplbiim 505 . . . . . . . . . 10 (∀𝑔𝑖𝑎 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅))
49 gonanegoal 32484 . . . . . . . . . . . . . . . 16 (𝑢𝑔𝑣) ≠ ∀𝑔𝑖𝑎
50 eqneqall 3031 . . . . . . . . . . . . . . . 16 ((𝑢𝑔𝑣) = ∀𝑔𝑖𝑎 → ((𝑢𝑔𝑣) ≠ ∀𝑔𝑖𝑎𝑎 ∈ (Fmla‘suc ∅)))
5149, 50mpi 20 . . . . . . . . . . . . . . 15 ((𝑢𝑔𝑣) = ∀𝑔𝑖𝑎𝑎 ∈ (Fmla‘suc ∅))
5251eqcoms 2832 . . . . . . . . . . . . . 14 (∀𝑔𝑖𝑎 = (𝑢𝑔𝑣) → 𝑎 ∈ (Fmla‘suc ∅))
5352a1i 11 . . . . . . . . . . . . 13 ((𝑢 ∈ (Fmla‘∅) ∧ 𝑣 ∈ (Fmla‘∅)) → (∀𝑔𝑖𝑎 = (𝑢𝑔𝑣) → 𝑎 ∈ (Fmla‘suc ∅)))
5453rexlimdva 3288 . . . . . . . . . . . 12 (𝑢 ∈ (Fmla‘∅) → (∃𝑣 ∈ (Fmla‘∅)∀𝑔𝑖𝑎 = (𝑢𝑔𝑣) → 𝑎 ∈ (Fmla‘suc ∅)))
55 df-goal 32474 . . . . . . . . . . . . . . 15 𝑔𝑘𝑢 = ⟨2o, ⟨𝑘, 𝑢⟩⟩
5625, 55eqeq12i 2839 . . . . . . . . . . . . . 14 (∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢 ↔ ⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨2o, ⟨𝑘, 𝑢⟩⟩)
5737, 38opth 5364 . . . . . . . . . . . . . . . . 17 (⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨2o, ⟨𝑘, 𝑢⟩⟩ ↔ (2o = 2o ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑢⟩))
58 vex 3502 . . . . . . . . . . . . . . . . . . 19 𝑖 ∈ V
59 vex 3502 . . . . . . . . . . . . . . . . . . 19 𝑎 ∈ V
6058, 59opth 5364 . . . . . . . . . . . . . . . . . 18 (⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑢⟩ ↔ (𝑖 = 𝑘𝑎 = 𝑢))
61 eleq1w 2899 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑎 → (𝑢 ∈ (Fmla‘∅) ↔ 𝑎 ∈ (Fmla‘∅)))
62 fmlasssuc 32521 . . . . . . . . . . . . . . . . . . . . . 22 (∅ ∈ ω → (Fmla‘∅) ⊆ (Fmla‘suc ∅))
6324, 62ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (Fmla‘∅) ⊆ (Fmla‘suc ∅)
6463sseli 3966 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅))
6561, 64syl6bi 254 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑎 → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
6665eqcoms 2832 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑢 → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
6760, 66simplbiim 505 . . . . . . . . . . . . . . . . 17 (⟨𝑖, 𝑎⟩ = ⟨𝑘, 𝑢⟩ → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
6857, 67simplbiim 505 . . . . . . . . . . . . . . . 16 (⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨2o, ⟨𝑘, 𝑢⟩⟩ → (𝑢 ∈ (Fmla‘∅) → 𝑎 ∈ (Fmla‘suc ∅)))
6968com12 32 . . . . . . . . . . . . . . 15 (𝑢 ∈ (Fmla‘∅) → (⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨2o, ⟨𝑘, 𝑢⟩⟩ → 𝑎 ∈ (Fmla‘suc ∅)))
7069adantr 481 . . . . . . . . . . . . . 14 ((𝑢 ∈ (Fmla‘∅) ∧ 𝑘 ∈ ω) → (⟨2o, ⟨𝑖, 𝑎⟩⟩ = ⟨2o, ⟨𝑘, 𝑢⟩⟩ → 𝑎 ∈ (Fmla‘suc ∅)))
7156, 70syl5bi 243 . . . . . . . . . . . . 13 ((𝑢 ∈ (Fmla‘∅) ∧ 𝑘 ∈ ω) → (∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢𝑎 ∈ (Fmla‘suc ∅)))
7271rexlimdva 3288 . . . . . . . . . . . 12 (𝑢 ∈ (Fmla‘∅) → (∃𝑘 ∈ ω ∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢𝑎 ∈ (Fmla‘suc ∅)))
7354, 72jaod 855 . . . . . . . . . . 11 (𝑢 ∈ (Fmla‘∅) → ((∃𝑣 ∈ (Fmla‘∅)∀𝑔𝑖𝑎 = (𝑢𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢) → 𝑎 ∈ (Fmla‘suc ∅)))
7473rexlimiv 3284 . . . . . . . . . 10 (∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀𝑔𝑖𝑎 = (𝑢𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢) → 𝑎 ∈ (Fmla‘suc ∅))
7548, 74jaoi 853 . . . . . . . . 9 ((∀𝑔𝑖𝑎 ∈ (Fmla‘∅) ∨ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)∀𝑔𝑖𝑎 = (𝑢𝑔𝑣) ∨ ∃𝑘 ∈ ω ∀𝑔𝑖𝑎 = ∀𝑔𝑘𝑢)) → 𝑎 ∈ (Fmla‘suc ∅))
7629, 75sylbi 218 . . . . . . . 8 (∀𝑔𝑖𝑎 ∈ (Fmla‘suc ∅) → 𝑎 ∈ (Fmla‘suc ∅))
77 goalrlem 32528 . . . . . . . 8 (𝑦 ∈ ω → ((∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑦) → 𝑎 ∈ (Fmla‘suc 𝑦)) → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑦) → 𝑎 ∈ (Fmla‘suc suc 𝑦))))
788, 13, 18, 23, 76, 77finds 7599 . . . . . . 7 (𝑛 ∈ ω → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛)))
7978adantr 481 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛)))
80 fveq2 6666 . . . . . . . . 9 (𝑁 = suc 𝑛 → (Fmla‘𝑁) = (Fmla‘suc 𝑛))
8180eleq2d 2902 . . . . . . . 8 (𝑁 = suc 𝑛 → (∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁) ↔ ∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛)))
8280eleq2d 2902 . . . . . . . 8 (𝑁 = suc 𝑛 → (𝑎 ∈ (Fmla‘𝑁) ↔ 𝑎 ∈ (Fmla‘suc 𝑛)))
8381, 82imbi12d 346 . . . . . . 7 (𝑁 = suc 𝑛 → ((∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛))))
8483adantl 482 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)) ↔ (∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑛) → 𝑎 ∈ (Fmla‘suc 𝑛))))
8579, 84mpbird 258 . . . . 5 ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → (∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)))
8685rexlimiva 3285 . . . 4 (∃𝑛 ∈ ω 𝑁 = suc 𝑛 → (∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)))
873, 86syl 17 . . 3 ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁) → 𝑎 ∈ (Fmla‘𝑁)))
8887impancom 452 . 2 ((𝑁 ∈ ω ∧ ∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → (𝑁 ≠ ∅ → 𝑎 ∈ (Fmla‘𝑁)))
892, 88mpd 15 1 ((𝑁 ∈ ω ∧ ∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → 𝑎 ∈ (Fmla‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wcel 2106  wne 3020  wrex 3143  Vcvv 3499  wss 3939  c0 4294  cop 4569  suc csuc 6190  cfv 6351  (class class class)co 7151  ωcom 7571  2oc2o 8090  𝑔cgoe 32465  𝑔cgna 32466  𝑔cgol 32467  Fmlacfmla 32469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-13 2385  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-map 8401  df-goel 32472  df-gona 32473  df-goal 32474  df-sat 32475  df-fmla 32477
This theorem is referenced by:  fmlasucdisj  32531
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