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Theorem nnsf 15051
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 6457 . . . . . . 7 1o ∈ 2o
32a1i 9 . . . . . 6 ((𝑝 ∈ ℕ𝑖 ∈ ω) → 1o ∈ 2o)
4 nninff 7135 . . . . . . . 8 (𝑝 ∈ ℕ𝑝:ω⟶2o)
54adantr 276 . . . . . . 7 ((𝑝 ∈ ℕ𝑖 ∈ ω) → 𝑝:ω⟶2o)
6 nnpredcl 4634 . . . . . . . 8 (𝑖 ∈ ω → 𝑖 ∈ ω)
76adantl 277 . . . . . . 7 ((𝑝 ∈ ℕ𝑖 ∈ ω) → 𝑖 ∈ ω)
85, 7ffvelcdmd 5665 . . . . . 6 ((𝑝 ∈ ℕ𝑖 ∈ ω) → (𝑝 𝑖) ∈ 2o)
9 nndceq0 4629 . . . . . . 7 (𝑖 ∈ ω → DECID 𝑖 = ∅)
109adantl 277 . . . . . 6 ((𝑝 ∈ ℕ𝑖 ∈ ω) → DECID 𝑖 = ∅)
113, 8, 10ifcldcd 3582 . . . . 5 ((𝑝 ∈ ℕ𝑖 ∈ ω) → if(𝑖 = ∅, 1o, (𝑝 𝑖)) ∈ 2o)
12 eqid 2187 . . . . 5 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))
1311, 12fmptd 5683 . . . 4 (𝑝 ∈ ℕ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))):ω⟶2o)
14 2onn 6536 . . . . 5 2o ∈ ω
15 omex 4604 . . . . 5 ω ∈ V
16 elmapg 6675 . . . . 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 426 . . . 4 ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) ∈ (2o𝑚 ω) ↔ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))):ω⟶2o)
1813, 17sylibr 134 . . 3 (𝑝 ∈ ℕ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) ∈ (2o𝑚 ω))
19 1on 6438 . . . . . . . . 9 1o ∈ On
2019ontrci 4439 . . . . . . . 8 Tr 1o
212a1i 9 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → 1o ∈ 2o)
224adantr 276 . . . . . . . . . . . 12 ((𝑝 ∈ ℕ𝑗 ∈ ω) → 𝑝:ω⟶2o)
23 peano2 4606 . . . . . . . . . . . . . 14 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
2423adantl 277 . . . . . . . . . . . . 13 ((𝑝 ∈ ℕ𝑗 ∈ ω) → suc 𝑗 ∈ ω)
25 nnpredcl 4634 . . . . . . . . . . . . 13 (suc 𝑗 ∈ ω → suc 𝑗 ∈ ω)
2624, 25syl 14 . . . . . . . . . . . 12 ((𝑝 ∈ ℕ𝑗 ∈ ω) → suc 𝑗 ∈ ω)
2722, 26ffvelcdmd 5665 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → (𝑝 suc 𝑗) ∈ 2o)
28 nndceq0 4629 . . . . . . . . . . . 12 (suc 𝑗 ∈ ω → DECID suc 𝑗 = ∅)
2924, 28syl 14 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → DECID suc 𝑗 = ∅)
3021, 27, 29ifcldcd 3582 . . . . . . . . . 10 ((𝑝 ∈ ℕ𝑗 ∈ ω) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ∈ 2o)
3130adantr 276 . . . . . . . . 9 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ∈ 2o)
32 df-2o 6432 . . . . . . . . 9 2o = suc 1o
3331, 32eleqtrdi 2280 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ∈ suc 1o)
34 trsucss 4435 . . . . . . . 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 3551 . . . . . . . 8 (𝑗 = ∅ → if(𝑗 = ∅, 1o, (𝑝 𝑗)) = 1o)
3736adantl 277 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(𝑗 = ∅, 1o, (𝑝 𝑗)) = 1o)
3835, 37sseqtrrd 3206 . . . . . 6 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝 𝑗)))
39 simpr 110 . . . . . . . . . . . 12 ((𝑝 ∈ ℕ𝑗 ∈ ω) → 𝑗 ∈ ω)
4039adantr 276 . . . . . . . . . . 11 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ∈ ω)
41 nnord 4623 . . . . . . . . . . 11 (𝑗 ∈ ω → Ord 𝑗)
42 ordtr 4390 . . . . . . . . . . 11 (Ord 𝑗 → Tr 𝑗)
4340, 41, 423syl 17 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → Tr 𝑗)
44 unisucg 4426 . . . . . . . . . . 11 (𝑗 ∈ ω → (Tr 𝑗 suc 𝑗 = 𝑗))
4540, 44syl 14 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (Tr 𝑗 suc 𝑗 = 𝑗))
4643, 45mpbid 147 . . . . . . . . 9 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → suc 𝑗 = 𝑗)
4746fveq2d 5531 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝 suc 𝑗) = (𝑝𝑗))
48 simpr 110 . . . . . . . . . . . 12 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ¬ 𝑗 = ∅)
4948neqned 2364 . . . . . . . . . . 11 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ≠ ∅)
50 nnsucpred 4628 . . . . . . . . . . 11 ((𝑗 ∈ ω ∧ 𝑗 ≠ ∅) → suc 𝑗 = 𝑗)
5140, 49, 50syl2anc 411 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → suc 𝑗 = 𝑗)
5251fveq2d 5531 . . . . . . . . 9 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘suc 𝑗) = (𝑝𝑗))
53 suceq 4414 . . . . . . . . . . . 12 (𝑘 = 𝑗 → suc 𝑘 = suc 𝑗)
5453fveq2d 5531 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝑝‘suc 𝑘) = (𝑝‘suc 𝑗))
55 fveq2 5527 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝑝𝑘) = (𝑝 𝑗))
5654, 55sseq12d 3198 . . . . . . . . . 10 (𝑘 = 𝑗 → ((𝑝‘suc 𝑘) ⊆ (𝑝𝑘) ↔ (𝑝‘suc 𝑗) ⊆ (𝑝 𝑗)))
57 fveq1 5526 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑝 → (𝑓‘suc 𝑗) = (𝑝‘suc 𝑗))
58 fveq1 5526 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑝 → (𝑓𝑗) = (𝑝𝑗))
5957, 58sseq12d 3198 . . . . . . . . . . . . . . 15 (𝑓 = 𝑝 → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ (𝑝‘suc 𝑗) ⊆ (𝑝𝑗)))
6059ralbidv 2487 . . . . . . . . . . . . . 14 (𝑓 = 𝑝 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝𝑗)))
61 df-nninf 7133 . . . . . . . . . . . . . 14 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
6260, 61elrab2 2908 . . . . . . . . . . . . 13 (𝑝 ∈ ℕ ↔ (𝑝 ∈ (2o𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝𝑗)))
6362simprbi 275 . . . . . . . . . . . 12 (𝑝 ∈ ℕ → ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝𝑗))
64 suceq 4414 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
6564fveq2d 5531 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝑝‘suc 𝑗) = (𝑝‘suc 𝑘))
66 fveq2 5527 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝑝𝑗) = (𝑝𝑘))
6765, 66sseq12d 3198 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → ((𝑝‘suc 𝑗) ⊆ (𝑝𝑗) ↔ (𝑝‘suc 𝑘) ⊆ (𝑝𝑘)))
6867cbvralv 2715 . . . . . . . . . . . 12 (∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝𝑗) ↔ ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝𝑘))
6963, 68sylib 122 . . . . . . . . . . 11 (𝑝 ∈ ℕ → ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝𝑘))
7069ad2antrr 488 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝𝑘))
71 nnpredcl 4634 . . . . . . . . . . . 12 (𝑗 ∈ ω → 𝑗 ∈ ω)
7271adantl 277 . . . . . . . . . . 11 ((𝑝 ∈ ℕ𝑗 ∈ ω) → 𝑗 ∈ ω)
7372adantr 276 . . . . . . . . . 10 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ∈ ω)
7456, 70, 73rspcdva 2858 . . . . . . . . 9 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘suc 𝑗) ⊆ (𝑝 𝑗))
7552, 74eqsstrrd 3204 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝𝑗) ⊆ (𝑝 𝑗))
7647, 75eqsstrd 3203 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝 suc 𝑗) ⊆ (𝑝 𝑗))
77 peano3 4607 . . . . . . . . . 10 (𝑗 ∈ ω → suc 𝑗 ≠ ∅)
7877neneqd 2378 . . . . . . . . 9 (𝑗 ∈ ω → ¬ suc 𝑗 = ∅)
7978ad2antlr 489 . . . . . . . 8 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ¬ suc 𝑗 = ∅)
8079iffalsed 3556 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) = (𝑝 suc 𝑗))
8148iffalsed 3556 . . . . . . 7 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(𝑗 = ∅, 1o, (𝑝 𝑗)) = (𝑝 𝑗))
8276, 80, 813sstr4d 3212 . . . . . 6 (((𝑝 ∈ ℕ𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝 𝑗)))
83 nndceq0 4629 . . . . . . . 8 (𝑗 ∈ ω → DECID 𝑗 = ∅)
8483adantl 277 . . . . . . 7 ((𝑝 ∈ ℕ𝑗 ∈ ω) → DECID 𝑗 = ∅)
85 exmiddc 837 . . . . . . 7 (DECID 𝑗 = ∅ → (𝑗 = ∅ ∨ ¬ 𝑗 = ∅))
8684, 85syl 14 . . . . . 6 ((𝑝 ∈ ℕ𝑗 ∈ ω) → (𝑗 = ∅ ∨ ¬ 𝑗 = ∅))
8738, 82, 86mpjaodan 799 . . . . 5 ((𝑝 ∈ ℕ𝑗 ∈ ω) → if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝 𝑗)))
88 eqeq1 2194 . . . . . . . 8 (𝑖 = suc 𝑗 → (𝑖 = ∅ ↔ suc 𝑗 = ∅))
89 unieq 3830 . . . . . . . . 9 (𝑖 = suc 𝑗 𝑖 = suc 𝑗)
9089fveq2d 5531 . . . . . . . 8 (𝑖 = suc 𝑗 → (𝑝 𝑖) = (𝑝 suc 𝑗))
9188, 90ifbieq2d 3570 . . . . . . 7 (𝑖 = suc 𝑗 → if(𝑖 = ∅, 1o, (𝑝 𝑖)) = if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)))
9291, 12fvmptg 5605 . . . . . 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 411 . . . . 5 ((𝑝 ∈ ℕ𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) = if(suc 𝑗 = ∅, 1o, (𝑝 suc 𝑗)))
9422, 72ffvelcdmd 5665 . . . . . . 7 ((𝑝 ∈ ℕ𝑗 ∈ ω) → (𝑝 𝑗) ∈ 2o)
9521, 94, 84ifcldcd 3582 . . . . . 6 ((𝑝 ∈ ℕ𝑗 ∈ ω) → if(𝑗 = ∅, 1o, (𝑝 𝑗)) ∈ 2o)
96 eqeq1 2194 . . . . . . . 8 (𝑖 = 𝑗 → (𝑖 = ∅ ↔ 𝑗 = ∅))
97 unieq 3830 . . . . . . . . 9 (𝑖 = 𝑗 𝑖 = 𝑗)
9897fveq2d 5531 . . . . . . . 8 (𝑖 = 𝑗 → (𝑝 𝑖) = (𝑝 𝑗))
9996, 98ifbieq2d 3570 . . . . . . 7 (𝑖 = 𝑗 → if(𝑖 = ∅, 1o, (𝑝 𝑖)) = if(𝑗 = ∅, 1o, (𝑝 𝑗)))
10099, 12fvmptg 5605 . . . . . 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 411 . . . . 5 ((𝑝 ∈ ℕ𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗) = if(𝑗 = ∅, 1o, (𝑝 𝑗)))
10287, 93, 1013sstr4d 3212 . . . 4 ((𝑝 ∈ ℕ𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗))
103102ralrimiva 2560 . . 3 (𝑝 ∈ ℕ → ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗))
104 fveq1 5526 . . . . . 6 (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘suc 𝑗))
105 fveq1 5526 . . . . . 6 (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) → (𝑓𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))‘𝑗))
106104, 105sseq12d 3198 . . . . 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 2487 . . . 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 2908 . . 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 417 . 2 (𝑝 ∈ ℕ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) ∈ ℕ)
1101, 109fmpti 5681 1 𝑆:ℕ⟶ℕ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wb 105  wo 709  DECID wdc 835   = wceq 1363  wcel 2158  wne 2357  wral 2465  Vcvv 2749  wss 3141  c0 3434  ifcif 3546   cuni 3821  cmpt 4076  Tr wtr 4113  Ord word 4374  suc csuc 4377  ωcom 4601  wf 5224  cfv 5228  (class class class)co 5888  1oc1o 6424  2oc2o 6425  𝑚 cmap 6662  xnninf 7132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1o 6431  df-2o 6432  df-map 6664  df-nninf 7133
This theorem is referenced by:  peano4nninf  15052
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