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Theorem nnsf 12783
Description: Domain and range of 𝑆. Part of Definition 3.3 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 30-Jul-2022.)
Hypothesis
Ref Expression
nns.s 𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))
Assertion
Ref Expression
nnsf 𝑆:ℕ⟶ℕ
Distinct variable group:   𝑖,𝑝
Allowed substitution hints:   𝑆(𝑖,𝑝)

Proof of Theorem nnsf
Dummy variables 𝑓 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nns.s . 2 𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))
2 1lt2o 6269 . . . . . . 7 1o ∈ 2o
32a1i 9 . . . . . 6 ((𝑝 ∈ ℕ𝑖 ∈ ω) → 1o ∈ 2o)
4 nninff 12782 . . . . . . . 8 (𝑝 ∈ ℕ𝑝:ω⟶2o)
54adantr 272 . . . . . . 7 ((𝑝 ∈ ℕ𝑖 ∈ ω) → 𝑝:ω⟶2o)
6 nnpredcl 4474 . . . . . . . 8 (𝑖 ∈ ω → 𝑖 ∈ ω)
76adantl 273 . . . . . . 7 ((𝑝 ∈ ℕ𝑖 ∈ ω) → 𝑖 ∈ ω)
85, 7ffvelrnd 5488 . . . . . 6 ((𝑝 ∈ ℕ𝑖 ∈ ω) → (𝑝 𝑖) ∈ 2o)
9 nndceq0 4469 . . . . . . 7 (𝑖 ∈ ω → DECID 𝑖 = ∅)
109adantl 273 . . . . . 6 ((𝑝 ∈ ℕ𝑖 ∈ ω) → DECID 𝑖 = ∅)
113, 8, 10ifcldcd 3454 . . . . 5 ((𝑝 ∈ ℕ𝑖 ∈ ω) → if(𝑖 = ∅, 1o, (𝑝 𝑖)) ∈ 2o)
12 eqid 2100 . . . . 5 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))
1311, 12fmptd 5506 . . . 4 (𝑝 ∈ ℕ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))):ω⟶2o)
14 2onn 6347 . . . . 5 2o ∈ ω
15 omex 4445 . . . . 5 ω ∈ V
16 elmapg 6485 . . . . 5 ((2o ∈ ω ∧ ω ∈ V) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) ∈ (2o𝑚 ω) ↔ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))):ω⟶2o))
1714, 15, 16mp2an 420 . . . 4 ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) ∈ (2o𝑚 ω) ↔ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))):ω⟶2o)
1813, 17sylibr 133 . . 3 (𝑝 ∈ ℕ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) ∈ (2o𝑚 ω))
19 1on 6250 . . . . . . . . 9 1o ∈ On
2019ontrci 4287 . . . . . . . 8 Tr 1o
212a1i 9 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → 1o ∈ 2o)
224adantr 272 . . . . . . . . . . . 12 ((𝑝 ∈ ℕ𝑗 ∈ ω) → 𝑝:ω⟶2o)
23 peano2 4447 . . . . . . . . . . . . . 14 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
2423adantl 273 . . . . . . . . . . . . 13 ((𝑝 ∈ ℕ𝑗 ∈ ω) → suc 𝑗 ∈ ω)
25 nnpredcl 4474 . . . . . . . . . . . . 13 (suc 𝑗 ∈ ω → suc 𝑗 ∈ ω)
2624, 25syl 14 . . . . . . . . . . . 12 ((𝑝 ∈ ℕ𝑗 ∈ ω) → suc 𝑗 ∈ ω)
2722, 26ffvelrnd 5488 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → (𝑝 suc 𝑗) ∈ 2o)
28 nndceq0 4469 . . . . . . . . . . . 12 (suc 𝑗 ∈ ω → DECID suc 𝑗 = ∅)
2924, 28syl 14 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → DECID suc 𝑗 = ∅)
3021, 27, 29ifcldcd 3454 . . . . . . . . . 10 ((𝑝 ∈ ℕ𝑗 ∈ ω) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ∈ 2o)
3130adantr 272 . . . . . . . . 9 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ∈ 2o)
32 df-2o 6244 . . . . . . . . 9 2o = suc 1o
3331, 32syl6eleq 2192 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ∈ suc 1o)
34 trsucss 4283 . . . . . . . 8 (Tr 1o → (if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ∈ suc 1o → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ⊆ 1o))
3520, 33, 34mpsyl 65 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ⊆ 1o)
36 iftrue 3426 . . . . . . . 8 (𝑗 = ∅ → if(𝑗 = ∅, 1o, (𝑝 𝑗)) = 1o)
3736adantl 273 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(𝑗 = ∅, 1o, (𝑝 𝑗)) = 1o)
3835, 37sseqtr4d 3086 . . . . . 6 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝 𝑗)))
39 simpr 109 . . . . . . . . . . . 12 ((𝑝 ∈ ℕ𝑗 ∈ ω) → 𝑗 ∈ ω)
4039adantr 272 . . . . . . . . . . 11 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ∈ ω)
41 nnord 4463 . . . . . . . . . . 11 (𝑗 ∈ ω → Ord 𝑗)
42 ordtr 4238 . . . . . . . . . . 11 (Ord 𝑗 → Tr 𝑗)
4340, 41, 423syl 17 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → Tr 𝑗)
44 unisucg 4274 . . . . . . . . . . 11 (𝑗 ∈ ω → (Tr 𝑗 suc 𝑗 = 𝑗))
4540, 44syl 14 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (Tr 𝑗 suc 𝑗 = 𝑗))
4643, 45mpbid 146 . . . . . . . . 9 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → suc 𝑗 = 𝑗)
4746fveq2d 5357 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝 suc 𝑗) = (𝑝𝑗))
48 simpr 109 . . . . . . . . . . . 12 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ¬ 𝑗 = ∅)
4948neqned 2274 . . . . . . . . . . 11 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ≠ ∅)
50 nnsucpred 4468 . . . . . . . . . . 11 ((𝑗 ∈ ω ∧ 𝑗 ≠ ∅) → suc 𝑗 = 𝑗)
5140, 49, 50syl2anc 406 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → suc 𝑗 = 𝑗)
5251fveq2d 5357 . . . . . . . . 9 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘suc 𝑗) = (𝑝𝑗))
53 suceq 4262 . . . . . . . . . . . 12 (𝑘 = 𝑗 → suc 𝑘 = suc 𝑗)
5453fveq2d 5357 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝑝‘suc 𝑘) = (𝑝‘suc 𝑗))
55 fveq2 5353 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝑝𝑘) = (𝑝 𝑗))
5654, 55sseq12d 3078 . . . . . . . . . 10 (𝑘 = 𝑗 → ((𝑝‘suc 𝑘) ⊆ (𝑝𝑘) ↔ (𝑝‘suc 𝑗) ⊆ (𝑝 𝑗)))
57 fveq1 5352 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑝 → (𝑓‘suc 𝑗) = (𝑝‘suc 𝑗))
58 fveq1 5352 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑝 → (𝑓𝑗) = (𝑝𝑗))
5957, 58sseq12d 3078 . . . . . . . . . . . . . . 15 (𝑓 = 𝑝 → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ (𝑝‘suc 𝑗) ⊆ (𝑝𝑗)))
6059ralbidv 2396 . . . . . . . . . . . . . 14 (𝑓 = 𝑝 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝𝑗)))
61 df-nninf 6919 . . . . . . . . . . . . . 14 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
6260, 61elrab2 2796 . . . . . . . . . . . . 13 (𝑝 ∈ ℕ ↔ (𝑝 ∈ (2o𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝𝑗)))
6362simprbi 271 . . . . . . . . . . . 12 (𝑝 ∈ ℕ → ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝𝑗))
64 suceq 4262 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
6564fveq2d 5357 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝑝‘suc 𝑗) = (𝑝‘suc 𝑘))
66 fveq2 5353 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝑝𝑗) = (𝑝𝑘))
6765, 66sseq12d 3078 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → ((𝑝‘suc 𝑗) ⊆ (𝑝𝑗) ↔ (𝑝‘suc 𝑘) ⊆ (𝑝𝑘)))
6867cbvralv 2612 . . . . . . . . . . . 12 (∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝𝑗) ↔ ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝𝑘))
6963, 68sylib 121 . . . . . . . . . . 11 (𝑝 ∈ ℕ → ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝𝑘))
7069ad2antrr 475 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝𝑘))
71 nnpredcl 4474 . . . . . . . . . . . 12 (𝑗 ∈ ω → 𝑗 ∈ ω)
7271adantl 273 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → 𝑗 ∈ ω)
7372adantr 272 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ∈ ω)
7456, 70, 73rspcdva 2749 . . . . . . . . 9 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘suc 𝑗) ⊆ (𝑝 𝑗))
7552, 74eqsstr3d 3084 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝𝑗) ⊆ (𝑝 𝑗))
7647, 75eqsstrd 3083 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝 suc 𝑗) ⊆ (𝑝 𝑗))
77 peano3 4448 . . . . . . . . . 10 (𝑗 ∈ ω → suc 𝑗 ≠ ∅)
7877neneqd 2288 . . . . . . . . 9 (𝑗 ∈ ω → ¬ suc 𝑗 = ∅)
7978ad2antlr 476 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ¬ suc 𝑗 = ∅)
8079iffalsed 3431 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) = (𝑝 suc 𝑗))
8148iffalsed 3431 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(𝑗 = ∅, 1o, (𝑝 𝑗)) = (𝑝 𝑗))
8276, 80, 813sstr4d 3092 . . . . . 6 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝 𝑗)))
83 nndceq0 4469 . . . . . . . 8 (𝑗 ∈ ω → DECID 𝑗 = ∅)
8483adantl 273 . . . . . . 7 ((𝑝 ∈ ℕ𝑗 ∈ ω) → DECID 𝑗 = ∅)
85 exmiddc 788 . . . . . . 7 (DECID 𝑗 = ∅ → (𝑗 = ∅ ∨ ¬ 𝑗 = ∅))
8684, 85syl 14 . . . . . 6 ((𝑝 ∈ ℕ𝑗 ∈ ω) → (𝑗 = ∅ ∨ ¬ 𝑗 = ∅))
8738, 82, 86mpjaodan 753 . . . . 5 ((𝑝 ∈ ℕ𝑗 ∈ ω) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝 𝑗)))
88 eqeq1 2106 . . . . . . . 8 (𝑖 = suc 𝑗 → (𝑖 = ∅ ↔ suc 𝑗 = ∅))
89 unieq 3692 . . . . . . . . 9 (𝑖 = suc 𝑗 𝑖 = suc 𝑗)
9089fveq2d 5357 . . . . . . . 8 (𝑖 = suc 𝑗 → (𝑝 𝑖) = (𝑝 suc 𝑗))
9188, 90ifbieq2d 3443 . . . . . . 7 (𝑖 = suc 𝑗 → if(𝑖 = ∅, 1o, (𝑝 𝑖)) = if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)))
9291, 12fvmptg 5429 . . . . . 6 ((suc 𝑗 ∈ ω ∧ if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ∈ 2o) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) = if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)))
9324, 30, 92syl2anc 406 . . . . 5 ((𝑝 ∈ ℕ𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) = if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)))
9422, 72ffvelrnd 5488 . . . . . . 7 ((𝑝 ∈ ℕ𝑗 ∈ ω) → (𝑝 𝑗) ∈ 2o)
9521, 94, 84ifcldcd 3454 . . . . . 6 ((𝑝 ∈ ℕ𝑗 ∈ ω) → if(𝑗 = ∅, 1o, (𝑝 𝑗)) ∈ 2o)
96 eqeq1 2106 . . . . . . . 8 (𝑖 = 𝑗 → (𝑖 = ∅ ↔ 𝑗 = ∅))
97 unieq 3692 . . . . . . . . 9 (𝑖 = 𝑗 𝑖 = 𝑗)
9897fveq2d 5357 . . . . . . . 8 (𝑖 = 𝑗 → (𝑝 𝑖) = (𝑝 𝑗))
9996, 98ifbieq2d 3443 . . . . . . 7 (𝑖 = 𝑗 → if(𝑖 = ∅, 1o, (𝑝 𝑖)) = if(𝑗 = ∅, 1o, (𝑝 𝑗)))
10099, 12fvmptg 5429 . . . . . 6 ((𝑗 ∈ ω ∧ if(𝑗 = ∅, 1o, (𝑝 𝑗)) ∈ 2o) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗) = if(𝑗 = ∅, 1o, (𝑝 𝑗)))
10139, 95, 100syl2anc 406 . . . . 5 ((𝑝 ∈ ℕ𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗) = if(𝑗 = ∅, 1o, (𝑝 𝑗)))
10287, 93, 1013sstr4d 3092 . . . 4 ((𝑝 ∈ ℕ𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗))
103102ralrimiva 2464 . . 3 (𝑝 ∈ ℕ → ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗))
104 fveq1 5352 . . . . . 6 (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗))
105 fveq1 5352 . . . . . 6 (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) → (𝑓𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗))
106104, 105sseq12d 3078 . . . . 5 (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗)))
107106ralbidv 2396 . . . 4 (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗)))
108107, 61elrab2 2796 . . 3 ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) ∈ ℕ ↔ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) ∈ (2o𝑚 ω) ∧ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗)))
10918, 103, 108sylanbrc 411 . 2 (𝑝 ∈ ℕ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) ∈ ℕ)
1101, 109fmpti 5504 1 𝑆:ℕ⟶ℕ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  wo 670  DECID wdc 786   = wceq 1299  wcel 1448  wne 2267  wral 2375  Vcvv 2641  wss 3021  c0 3310  ifcif 3421   cuni 3683  cmpt 3929  Tr wtr 3966  Ord word 4222  suc csuc 4225  ωcom 4442  wf 5055  cfv 5059  (class class class)co 5706  1oc1o 6236  2oc2o 6237  𝑚 cmap 6472  xnninf 6917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1o 6243  df-2o 6244  df-map 6474  df-nninf 6919
This theorem is referenced by:  peano4nninf  12784
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