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Theorem nnsf 13737
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 6402 . . . . . . 7 1o ∈ 2o
32a1i 9 . . . . . 6 ((𝑝 ∈ ℕ𝑖 ∈ ω) → 1o ∈ 2o)
4 nninff 7079 . . . . . . . 8 (𝑝 ∈ ℕ𝑝:ω⟶2o)
54adantr 274 . . . . . . 7 ((𝑝 ∈ ℕ𝑖 ∈ ω) → 𝑝:ω⟶2o)
6 nnpredcl 4595 . . . . . . . 8 (𝑖 ∈ ω → 𝑖 ∈ ω)
76adantl 275 . . . . . . 7 ((𝑝 ∈ ℕ𝑖 ∈ ω) → 𝑖 ∈ ω)
85, 7ffvelrnd 5616 . . . . . 6 ((𝑝 ∈ ℕ𝑖 ∈ ω) → (𝑝 𝑖) ∈ 2o)
9 nndceq0 4590 . . . . . . 7 (𝑖 ∈ ω → DECID 𝑖 = ∅)
109adantl 275 . . . . . 6 ((𝑝 ∈ ℕ𝑖 ∈ ω) → DECID 𝑖 = ∅)
113, 8, 10ifcldcd 3551 . . . . 5 ((𝑝 ∈ ℕ𝑖 ∈ ω) → if(𝑖 = ∅, 1o, (𝑝 𝑖)) ∈ 2o)
12 eqid 2164 . . . . 5 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))
1311, 12fmptd 5634 . . . 4 (𝑝 ∈ ℕ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))):ω⟶2o)
14 2onn 6481 . . . . 5 2o ∈ ω
15 omex 4565 . . . . 5 ω ∈ V
16 elmapg 6619 . . . . 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 423 . . . 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 6383 . . . . . . . . 9 1o ∈ On
2019ontrci 4400 . . . . . . . 8 Tr 1o
212a1i 9 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → 1o ∈ 2o)
224adantr 274 . . . . . . . . . . . 12 ((𝑝 ∈ ℕ𝑗 ∈ ω) → 𝑝:ω⟶2o)
23 peano2 4567 . . . . . . . . . . . . . 14 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
2423adantl 275 . . . . . . . . . . . . 13 ((𝑝 ∈ ℕ𝑗 ∈ ω) → suc 𝑗 ∈ ω)
25 nnpredcl 4595 . . . . . . . . . . . . 13 (suc 𝑗 ∈ ω → suc 𝑗 ∈ ω)
2624, 25syl 14 . . . . . . . . . . . 12 ((𝑝 ∈ ℕ𝑗 ∈ ω) → suc 𝑗 ∈ ω)
2722, 26ffvelrnd 5616 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → (𝑝 suc 𝑗) ∈ 2o)
28 nndceq0 4590 . . . . . . . . . . . 12 (suc 𝑗 ∈ ω → DECID suc 𝑗 = ∅)
2924, 28syl 14 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → DECID suc 𝑗 = ∅)
3021, 27, 29ifcldcd 3551 . . . . . . . . . 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 6377 . . . . . . . . 9 2o = suc 1o
3331, 32eleqtrdi 2257 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ∈ suc 1o)
34 trsucss 4396 . . . . . . . 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 3521 . . . . . . . 8 (𝑗 = ∅ → if(𝑗 = ∅, 1o, (𝑝 𝑗)) = 1o)
3736adantl 275 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(𝑗 = ∅, 1o, (𝑝 𝑗)) = 1o)
3835, 37sseqtrrd 3177 . . . . . 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 4584 . . . . . . . . . . 11 (𝑗 ∈ ω → Ord 𝑗)
42 ordtr 4351 . . . . . . . . . . 11 (Ord 𝑗 → Tr 𝑗)
4340, 41, 423syl 17 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → Tr 𝑗)
44 unisucg 4387 . . . . . . . . . . 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 5485 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝 suc 𝑗) = (𝑝𝑗))
48 simpr 109 . . . . . . . . . . . 12 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ¬ 𝑗 = ∅)
4948neqned 2341 . . . . . . . . . . 11 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ≠ ∅)
50 nnsucpred 4589 . . . . . . . . . . 11 ((𝑗 ∈ ω ∧ 𝑗 ≠ ∅) → suc 𝑗 = 𝑗)
5140, 49, 50syl2anc 409 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → suc 𝑗 = 𝑗)
5251fveq2d 5485 . . . . . . . . 9 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘suc 𝑗) = (𝑝𝑗))
53 suceq 4375 . . . . . . . . . . . 12 (𝑘 = 𝑗 → suc 𝑘 = suc 𝑗)
5453fveq2d 5485 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝑝‘suc 𝑘) = (𝑝‘suc 𝑗))
55 fveq2 5481 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝑝𝑘) = (𝑝 𝑗))
5654, 55sseq12d 3169 . . . . . . . . . 10 (𝑘 = 𝑗 → ((𝑝‘suc 𝑘) ⊆ (𝑝𝑘) ↔ (𝑝‘suc 𝑗) ⊆ (𝑝 𝑗)))
57 fveq1 5480 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑝 → (𝑓‘suc 𝑗) = (𝑝‘suc 𝑗))
58 fveq1 5480 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑝 → (𝑓𝑗) = (𝑝𝑗))
5957, 58sseq12d 3169 . . . . . . . . . . . . . . 15 (𝑓 = 𝑝 → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ (𝑝‘suc 𝑗) ⊆ (𝑝𝑗)))
6059ralbidv 2464 . . . . . . . . . . . . . 14 (𝑓 = 𝑝 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝𝑗)))
61 df-nninf 7077 . . . . . . . . . . . . . 14 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
6260, 61elrab2 2881 . . . . . . . . . . . . 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 4375 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
6564fveq2d 5485 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝑝‘suc 𝑗) = (𝑝‘suc 𝑘))
66 fveq2 5481 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝑝𝑗) = (𝑝𝑘))
6765, 66sseq12d 3169 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → ((𝑝‘suc 𝑗) ⊆ (𝑝𝑗) ↔ (𝑝‘suc 𝑘) ⊆ (𝑝𝑘)))
6867cbvralv 2690 . . . . . . . . . . . 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 480 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝𝑘))
71 nnpredcl 4595 . . . . . . . . . . . 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 2831 . . . . . . . . 9 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘suc 𝑗) ⊆ (𝑝 𝑗))
7552, 74eqsstrrd 3175 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝𝑗) ⊆ (𝑝 𝑗))
7647, 75eqsstrd 3174 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝 suc 𝑗) ⊆ (𝑝 𝑗))
77 peano3 4568 . . . . . . . . . 10 (𝑗 ∈ ω → suc 𝑗 ≠ ∅)
7877neneqd 2355 . . . . . . . . 9 (𝑗 ∈ ω → ¬ suc 𝑗 = ∅)
7978ad2antlr 481 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ¬ suc 𝑗 = ∅)
8079iffalsed 3526 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) = (𝑝 suc 𝑗))
8148iffalsed 3526 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(𝑗 = ∅, 1o, (𝑝 𝑗)) = (𝑝 𝑗))
8276, 80, 813sstr4d 3183 . . . . . 6 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝 𝑗)))
83 nndceq0 4590 . . . . . . . 8 (𝑗 ∈ ω → DECID 𝑗 = ∅)
8483adantl 275 . . . . . . 7 ((𝑝 ∈ ℕ𝑗 ∈ ω) → DECID 𝑗 = ∅)
85 exmiddc 826 . . . . . . 7 (DECID 𝑗 = ∅ → (𝑗 = ∅ ∨ ¬ 𝑗 = ∅))
8684, 85syl 14 . . . . . 6 ((𝑝 ∈ ℕ𝑗 ∈ ω) → (𝑗 = ∅ ∨ ¬ 𝑗 = ∅))
8738, 82, 86mpjaodan 788 . . . . 5 ((𝑝 ∈ ℕ𝑗 ∈ ω) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝 𝑗)))
88 eqeq1 2171 . . . . . . . 8 (𝑖 = suc 𝑗 → (𝑖 = ∅ ↔ suc 𝑗 = ∅))
89 unieq 3793 . . . . . . . . 9 (𝑖 = suc 𝑗 𝑖 = suc 𝑗)
9089fveq2d 5485 . . . . . . . 8 (𝑖 = suc 𝑗 → (𝑝 𝑖) = (𝑝 suc 𝑗))
9188, 90ifbieq2d 3540 . . . . . . 7 (𝑖 = suc 𝑗 → if(𝑖 = ∅, 1o, (𝑝 𝑖)) = if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)))
9291, 12fvmptg 5557 . . . . . 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 409 . . . . 5 ((𝑝 ∈ ℕ𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) = if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)))
9422, 72ffvelrnd 5616 . . . . . . 7 ((𝑝 ∈ ℕ𝑗 ∈ ω) → (𝑝 𝑗) ∈ 2o)
9521, 94, 84ifcldcd 3551 . . . . . 6 ((𝑝 ∈ ℕ𝑗 ∈ ω) → if(𝑗 = ∅, 1o, (𝑝 𝑗)) ∈ 2o)
96 eqeq1 2171 . . . . . . . 8 (𝑖 = 𝑗 → (𝑖 = ∅ ↔ 𝑗 = ∅))
97 unieq 3793 . . . . . . . . 9 (𝑖 = 𝑗 𝑖 = 𝑗)
9897fveq2d 5485 . . . . . . . 8 (𝑖 = 𝑗 → (𝑝 𝑖) = (𝑝 𝑗))
9996, 98ifbieq2d 3540 . . . . . . 7 (𝑖 = 𝑗 → if(𝑖 = ∅, 1o, (𝑝 𝑖)) = if(𝑗 = ∅, 1o, (𝑝 𝑗)))
10099, 12fvmptg 5557 . . . . . 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 409 . . . . 5 ((𝑝 ∈ ℕ𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗) = if(𝑗 = ∅, 1o, (𝑝 𝑗)))
10287, 93, 1013sstr4d 3183 . . . 4 ((𝑝 ∈ ℕ𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗))
103102ralrimiva 2537 . . 3 (𝑝 ∈ ℕ → ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗))
104 fveq1 5480 . . . . . 6 (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗))
105 fveq1 5480 . . . . . 6 (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) → (𝑓𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗))
106104, 105sseq12d 3169 . . . . 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 2464 . . . 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 2881 . . 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 414 . 2 (𝑝 ∈ ℕ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) ∈ ℕ)
1101, 109fmpti 5632 1 𝑆:ℕ⟶ℕ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  wo 698  DECID wdc 824   = wceq 1342  wcel 2135  wne 2334  wral 2442  Vcvv 2722  wss 3112  c0 3405  ifcif 3516   cuni 3784  cmpt 4038  Tr wtr 4075  Ord word 4335  suc csuc 4338  ωcom 4562  wf 5179  cfv 5183  (class class class)co 5837  1oc1o 6369  2oc2o 6370  𝑚 cmap 6606  xnninf 7076
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-iinf 4560
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2724  df-sbc 2948  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-if 3517  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-int 3820  df-br 3978  df-opab 4039  df-mpt 4040  df-tr 4076  df-id 4266  df-iord 4339  df-on 4341  df-suc 4344  df-iom 4563  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-fv 5191  df-ov 5840  df-oprab 5841  df-mpo 5842  df-1o 6376  df-2o 6377  df-map 6608  df-nninf 7077
This theorem is referenced by:  peano4nninf  13738
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