Theorem n0sind 27942

Description: Principle of Mathematical Induction (inference schema). Compare nnind 12234 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

Step	Hyp	Ref	Expression
1		0n0s 27939	. . . . . 6 ⊢ 0s ∈ ℕ0s
2		n0sind.5	. . . . . 6 ⊢ 𝜓
3		n0sind.1	. . . . . . 7 ⊢ (𝑥 = 0s → (𝜑 ↔ 𝜓))
4	3	elrab 3682	. . . . . 6 ⊢ ( 0s ∈ {𝑥 ∈ ℕ0s ∣ 𝜑} ↔ ( 0s ∈ ℕ0s ∧ 𝜓))
5	1, 2, 4	mpbir2an 707	. . . . 5 ⊢ 0s ∈ {𝑥 ∈ ℕ0s ∣ 𝜑}
6		n0sind.6	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → (𝜒 → 𝜃))
7		peano2n0s 27940	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
8	6, 7	jctild 524	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ0s → (𝜒 → ((𝑦 +s 1s ) ∈ ℕ0s ∧ 𝜃)))
9	8	imp 405	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜒) → ((𝑦 +s 1s ) ∈ ℕ0s ∧ 𝜃))
10		n0sind.2	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
11	10	elrab 3682	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ0s ∣ 𝜑} ↔ (𝑦 ∈ ℕ0s ∧ 𝜒))
12		n0sind.3	. . . . . . . 8 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝜑 ↔ 𝜃))
13	12	elrab 3682	. . . . . . 7 ⊢ ((𝑦 +s 1s ) ∈ {𝑥 ∈ ℕ0s ∣ 𝜑} ↔ ((𝑦 +s 1s ) ∈ ℕ0s ∧ 𝜃))
14	9, 11, 13	3imtr4i 291	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ0s ∣ 𝜑} → (𝑦 +s 1s ) ∈ {𝑥 ∈ ℕ0s ∣ 𝜑})
15	14	rgen 3061	. . . . 5 ⊢ ∀𝑦 ∈ {𝑥 ∈ ℕ0s ∣ 𝜑} (𝑦 +s 1s ) ∈ {𝑥 ∈ ℕ0s ∣ 𝜑}
16		peano5n0s 27935	. . . . 5 ⊢ (( 0s ∈ {𝑥 ∈ ℕ0s ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ0s ∣ 𝜑} (𝑦 +s 1s ) ∈ {𝑥 ∈ ℕ0s ∣ 𝜑}) → ℕ0s ⊆ {𝑥 ∈ ℕ0s ∣ 𝜑})
17	5, 15, 16	mp2an 688	. . . 4 ⊢ ℕ0s ⊆ {𝑥 ∈ ℕ0s ∣ 𝜑}
18	17	sseli 3977	. . 3 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ {𝑥 ∈ ℕ0s ∣ 𝜑})
19		n0sind.4	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
20	19	elrab 3682	. . 3 ⊢ (𝐴 ∈ {𝑥 ∈ ℕ0s ∣ 𝜑} ↔ (𝐴 ∈ ℕ0s ∧ 𝜏))
21	18, 20	sylib 217	. 2 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ∈ ℕ0s ∧ 𝜏))
22	21	simprd 494	1 ⊢ (𝐴 ∈ ℕ0s → 𝜏)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059  {crab 3430   ⊆ wss 3947  (class class class)co 7411   0_s c0s 27560   1_s c1s 27561   +_s cadds 27681  ℕ_0scnn0s 27929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-no 27382  df-slt 27383  df-bday 27384  df-sslt 27519  df-scut 27521  df-0s 27562  df-n0s 27931

This theorem is referenced by:  n0scut  27943