Theorem n0sind 28599

Description: Principle of Mathematical Induction (inference schema). Compare nnind 12278 and finds 7896. (Contributed by Scott Fenton, 17-Mar-2025.)

Hypotheses

Ref	Expression
n0sind.1	⊢ (𝑥 = 0s → (𝜑 ↔ 𝜓))
n0sind.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
n0sind.3	⊢ (𝑥 = (𝑦 +s 1s ) → (𝜑 ↔ 𝜃))
n0sind.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
n0sind.5	⊢ 𝜓
n0sind.6	⊢ (𝑦 ∈ ℕ0s → (𝜒 → 𝜃))

Assertion

Ref	Expression
n0sind	⊢ (𝐴 ∈ ℕ0s → 𝜏)

Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem n0sind

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tru 1574	. 2 ⊢ ⊤
2		df-n0s 28580	. . . 4 ⊢ ℕ0s = (rec((𝑛 ∈ V ↦ (𝑛 +s 1s )), 0s ) “ ω)
3	2	a1i 11	. . 3 ⊢ (⊤ → ℕ0s = (rec((𝑛 ∈ V ↦ (𝑛 +s 1s )), 0s ) “ ω))
4		0no 28075	. . . 4 ⊢ 0s ∈ No
5	4	a1i 11	. . 3 ⊢ (⊤ → 0s ∈ No )
6		n0sind.1	. . 3 ⊢ (𝑥 = 0s → (𝜑 ↔ 𝜓))
7		n0sind.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
8		n0sind.3	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝜑 ↔ 𝜃))
9		n0sind.4	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
10		n0sind.5	. . . 4 ⊢ 𝜓
11	10	a1i 11	. . 3 ⊢ (⊤ → 𝜓)
12		n0sind.6	. . . 4 ⊢ (𝑦 ∈ ℕ0s → (𝜒 → 𝜃))
13	12	adantl 487	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℕ0s) → (𝜒 → 𝜃))
14	3, 5, 6, 7, 8, 9, 11, 13	noseqinds 28559	. 2 ⊢ ((⊤ ∧ 𝐴 ∈ ℕ0s) → 𝜏)
15	1, 14	mpan 703	1 ⊢ (𝐴 ∈ ℕ0s → 𝜏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145  Vcvv 3453   ↦ cmpt 5190   “ cima 5662  (class class class)co 7416  ωcom 7865  reccrdg 8401   No csur 27877   0_s c0s 28071   1_s c1s 28072   +_s cadds 28225  ℕ_0scn0s 28578

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-no 27880  df-lts 27881  df-bday 27882  df-slts 28024  df-cuts 28026  df-0s 28073  df-n0s 28580

This theorem is used by:  n0cut  28600  n0sge0  28604  n0s0suc  28608  n0addscl  28610  n0mulscl  28611  n0bday  28618  n0s0m1  28628  n0subs  28629  n0p1nns  28637  dfnns2  28638  eucliddivs  28642  peano5uzs  28670  n0seo  28687  expscllem  28696  expadds  28701  expsne0  28702  expsgt0  28703  pw2recs  28704  pw2cut  28726  pw2cut2  28728  bdaypw2n0bndlem  28729  bdayfinbndlem2  28734  z12zsodd  28748