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Theorem nnsf 13285
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 6339 . . . . . . 7 1o ∈ 2o
32a1i 9 . . . . . 6 ((𝑝 ∈ ℕ𝑖 ∈ ω) → 1o ∈ 2o)
4 nninff 13284 . . . . . . . 8 (𝑝 ∈ ℕ𝑝:ω⟶2o)
54adantr 274 . . . . . . 7 ((𝑝 ∈ ℕ𝑖 ∈ ω) → 𝑝:ω⟶2o)
6 nnpredcl 4536 . . . . . . . 8 (𝑖 ∈ ω → 𝑖 ∈ ω)
76adantl 275 . . . . . . 7 ((𝑝 ∈ ℕ𝑖 ∈ ω) → 𝑖 ∈ ω)
85, 7ffvelrnd 5556 . . . . . 6 ((𝑝 ∈ ℕ𝑖 ∈ ω) → (𝑝 𝑖) ∈ 2o)
9 nndceq0 4531 . . . . . . 7 (𝑖 ∈ ω → DECID 𝑖 = ∅)
109adantl 275 . . . . . 6 ((𝑝 ∈ ℕ𝑖 ∈ ω) → DECID 𝑖 = ∅)
113, 8, 10ifcldcd 3507 . . . . 5 ((𝑝 ∈ ℕ𝑖 ∈ ω) → if(𝑖 = ∅, 1o, (𝑝 𝑖)) ∈ 2o)
12 eqid 2139 . . . . 5 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))
1311, 12fmptd 5574 . . . 4 (𝑝 ∈ ℕ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))):ω⟶2o)
14 2onn 6417 . . . . 5 2o ∈ ω
15 omex 4507 . . . . 5 ω ∈ V
16 elmapg 6555 . . . . 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 422 . . . 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 6320 . . . . . . . . 9 1o ∈ On
2019ontrci 4349 . . . . . . . 8 Tr 1o
212a1i 9 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → 1o ∈ 2o)
224adantr 274 . . . . . . . . . . . 12 ((𝑝 ∈ ℕ𝑗 ∈ ω) → 𝑝:ω⟶2o)
23 peano2 4509 . . . . . . . . . . . . . 14 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
2423adantl 275 . . . . . . . . . . . . 13 ((𝑝 ∈ ℕ𝑗 ∈ ω) → suc 𝑗 ∈ ω)
25 nnpredcl 4536 . . . . . . . . . . . . 13 (suc 𝑗 ∈ ω → suc 𝑗 ∈ ω)
2624, 25syl 14 . . . . . . . . . . . 12 ((𝑝 ∈ ℕ𝑗 ∈ ω) → suc 𝑗 ∈ ω)
2722, 26ffvelrnd 5556 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → (𝑝 suc 𝑗) ∈ 2o)
28 nndceq0 4531 . . . . . . . . . . . 12 (suc 𝑗 ∈ ω → DECID suc 𝑗 = ∅)
2924, 28syl 14 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → DECID suc 𝑗 = ∅)
3021, 27, 29ifcldcd 3507 . . . . . . . . . 10 ((𝑝 ∈ ℕ𝑗 ∈ ω) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ∈ 2o)
3130adantr 274 . . . . . . . . 9 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ∈ 2o)
32 df-2o 6314 . . . . . . . . 9 2o = suc 1o
3331, 32eleqtrdi 2232 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ∈ suc 1o)
34 trsucss 4345 . . . . . . . 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 3479 . . . . . . . 8 (𝑗 = ∅ → if(𝑗 = ∅, 1o, (𝑝 𝑗)) = 1o)
3736adantl 275 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(𝑗 = ∅, 1o, (𝑝 𝑗)) = 1o)
3835, 37sseqtrrd 3136 . . . . . 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 274 . . . . . . . . . . 11 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ∈ ω)
41 nnord 4525 . . . . . . . . . . 11 (𝑗 ∈ ω → Ord 𝑗)
42 ordtr 4300 . . . . . . . . . . 11 (Ord 𝑗 → Tr 𝑗)
4340, 41, 423syl 17 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → Tr 𝑗)
44 unisucg 4336 . . . . . . . . . . 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 5425 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝 suc 𝑗) = (𝑝𝑗))
48 simpr 109 . . . . . . . . . . . 12 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ¬ 𝑗 = ∅)
4948neqned 2315 . . . . . . . . . . 11 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ≠ ∅)
50 nnsucpred 4530 . . . . . . . . . . 11 ((𝑗 ∈ ω ∧ 𝑗 ≠ ∅) → suc 𝑗 = 𝑗)
5140, 49, 50syl2anc 408 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → suc 𝑗 = 𝑗)
5251fveq2d 5425 . . . . . . . . 9 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘suc 𝑗) = (𝑝𝑗))
53 suceq 4324 . . . . . . . . . . . 12 (𝑘 = 𝑗 → suc 𝑘 = suc 𝑗)
5453fveq2d 5425 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝑝‘suc 𝑘) = (𝑝‘suc 𝑗))
55 fveq2 5421 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝑝𝑘) = (𝑝 𝑗))
5654, 55sseq12d 3128 . . . . . . . . . 10 (𝑘 = 𝑗 → ((𝑝‘suc 𝑘) ⊆ (𝑝𝑘) ↔ (𝑝‘suc 𝑗) ⊆ (𝑝 𝑗)))
57 fveq1 5420 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑝 → (𝑓‘suc 𝑗) = (𝑝‘suc 𝑗))
58 fveq1 5420 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑝 → (𝑓𝑗) = (𝑝𝑗))
5957, 58sseq12d 3128 . . . . . . . . . . . . . . 15 (𝑓 = 𝑝 → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ (𝑝‘suc 𝑗) ⊆ (𝑝𝑗)))
6059ralbidv 2437 . . . . . . . . . . . . . 14 (𝑓 = 𝑝 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝𝑗)))
61 df-nninf 7007 . . . . . . . . . . . . . 14 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
6260, 61elrab2 2843 . . . . . . . . . . . . 13 (𝑝 ∈ ℕ ↔ (𝑝 ∈ (2o𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝𝑗)))
6362simprbi 273 . . . . . . . . . . . 12 (𝑝 ∈ ℕ → ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝𝑗))
64 suceq 4324 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
6564fveq2d 5425 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝑝‘suc 𝑗) = (𝑝‘suc 𝑘))
66 fveq2 5421 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝑝𝑗) = (𝑝𝑘))
6765, 66sseq12d 3128 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → ((𝑝‘suc 𝑗) ⊆ (𝑝𝑗) ↔ (𝑝‘suc 𝑘) ⊆ (𝑝𝑘)))
6867cbvralv 2654 . . . . . . . . . . . 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 479 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝𝑘))
71 nnpredcl 4536 . . . . . . . . . . . 12 (𝑗 ∈ ω → 𝑗 ∈ ω)
7271adantl 275 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → 𝑗 ∈ ω)
7372adantr 274 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ∈ ω)
7456, 70, 73rspcdva 2794 . . . . . . . . 9 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘suc 𝑗) ⊆ (𝑝 𝑗))
7552, 74eqsstrrd 3134 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝𝑗) ⊆ (𝑝 𝑗))
7647, 75eqsstrd 3133 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝 suc 𝑗) ⊆ (𝑝 𝑗))
77 peano3 4510 . . . . . . . . . 10 (𝑗 ∈ ω → suc 𝑗 ≠ ∅)
7877neneqd 2329 . . . . . . . . 9 (𝑗 ∈ ω → ¬ suc 𝑗 = ∅)
7978ad2antlr 480 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ¬ suc 𝑗 = ∅)
8079iffalsed 3484 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) = (𝑝 suc 𝑗))
8148iffalsed 3484 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(𝑗 = ∅, 1o, (𝑝 𝑗)) = (𝑝 𝑗))
8276, 80, 813sstr4d 3142 . . . . . 6 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝 𝑗)))
83 nndceq0 4531 . . . . . . . 8 (𝑗 ∈ ω → DECID 𝑗 = ∅)
8483adantl 275 . . . . . . 7 ((𝑝 ∈ ℕ𝑗 ∈ ω) → DECID 𝑗 = ∅)
85 exmiddc 821 . . . . . . 7 (DECID 𝑗 = ∅ → (𝑗 = ∅ ∨ ¬ 𝑗 = ∅))
8684, 85syl 14 . . . . . 6 ((𝑝 ∈ ℕ𝑗 ∈ ω) → (𝑗 = ∅ ∨ ¬ 𝑗 = ∅))
8738, 82, 86mpjaodan 787 . . . . 5 ((𝑝 ∈ ℕ𝑗 ∈ ω) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝 𝑗)))
88 eqeq1 2146 . . . . . . . 8 (𝑖 = suc 𝑗 → (𝑖 = ∅ ↔ suc 𝑗 = ∅))
89 unieq 3745 . . . . . . . . 9 (𝑖 = suc 𝑗 𝑖 = suc 𝑗)
9089fveq2d 5425 . . . . . . . 8 (𝑖 = suc 𝑗 → (𝑝 𝑖) = (𝑝 suc 𝑗))
9188, 90ifbieq2d 3496 . . . . . . 7 (𝑖 = suc 𝑗 → if(𝑖 = ∅, 1o, (𝑝 𝑖)) = if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)))
9291, 12fvmptg 5497 . . . . . 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 408 . . . . 5 ((𝑝 ∈ ℕ𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) = if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)))
9422, 72ffvelrnd 5556 . . . . . . 7 ((𝑝 ∈ ℕ𝑗 ∈ ω) → (𝑝 𝑗) ∈ 2o)
9521, 94, 84ifcldcd 3507 . . . . . 6 ((𝑝 ∈ ℕ𝑗 ∈ ω) → if(𝑗 = ∅, 1o, (𝑝 𝑗)) ∈ 2o)
96 eqeq1 2146 . . . . . . . 8 (𝑖 = 𝑗 → (𝑖 = ∅ ↔ 𝑗 = ∅))
97 unieq 3745 . . . . . . . . 9 (𝑖 = 𝑗 𝑖 = 𝑗)
9897fveq2d 5425 . . . . . . . 8 (𝑖 = 𝑗 → (𝑝 𝑖) = (𝑝 𝑗))
9996, 98ifbieq2d 3496 . . . . . . 7 (𝑖 = 𝑗 → if(𝑖 = ∅, 1o, (𝑝 𝑖)) = if(𝑗 = ∅, 1o, (𝑝 𝑗)))
10099, 12fvmptg 5497 . . . . . 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 408 . . . . 5 ((𝑝 ∈ ℕ𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗) = if(𝑗 = ∅, 1o, (𝑝 𝑗)))
10287, 93, 1013sstr4d 3142 . . . 4 ((𝑝 ∈ ℕ𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗))
103102ralrimiva 2505 . . 3 (𝑝 ∈ ℕ → ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗))
104 fveq1 5420 . . . . . 6 (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗))
105 fveq1 5420 . . . . . 6 (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) → (𝑓𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗))
106104, 105sseq12d 3128 . . . . 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 2437 . . . 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 2843 . . 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 413 . 2 (𝑝 ∈ ℕ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) ∈ ℕ)
1101, 109fmpti 5572 1 𝑆:ℕ⟶ℕ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  wo 697  DECID wdc 819   = wceq 1331  wcel 1480  wne 2308  wral 2416  Vcvv 2686  wss 3071  c0 3363  ifcif 3474   cuni 3736  cmpt 3989  Tr wtr 4026  Ord word 4284  suc csuc 4287  ωcom 4504  wf 5119  cfv 5123  (class class class)co 5774  1oc1o 6306  2oc2o 6307  𝑚 cmap 6542  xnninf 7005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1o 6313  df-2o 6314  df-map 6544  df-nninf 7007
This theorem is referenced by:  peano4nninf  13286
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