Theorem n0sind 28653

Description: Principle of Mathematical Induction (inference schema). Compare nnind 12323 and finds 7891. (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 28634	. . . 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 28129	. . . 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 28613	. 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 3450   ↦ cmpt 5185   “ cima 5650  (class class class)co 7408  ωcom 7860  reccrdg 8395   No csur 27931   0_s c0s 28125   1_s c1s 28126   +_s cadds 28279  ℕ_0scn0s 28632

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 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-no 27934  df-lts 27935  df-bday 27936  df-slts 28078  df-cuts 28080  df-0s 28127  df-n0s 28634

This theorem is used by:  n0cut  28654  n0sge0  28658  n0s0suc  28662  n0addscl  28664  n0mulscl  28665  n0bday  28672  n0s0m1  28682  n0subs  28683  n0p1nns  28691  dfnns2  28692  eucliddivs  28696  peano5uzs  28724  n0seo  28741  expscllem  28750  expadds  28755  expsne0  28756  expsgt0  28757  pw2recs  28758  pw2cut  28780  pw2cut2  28782  bdaypw2n0bndlem  28783  bdayfinbndlem2  28788  z12zsodd  28802