Theorem noseqinds 28216

Description: Induction schema for surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)

Hypotheses

Ref	Expression
noseq.1	⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
noseq.2	⊢ (𝜑 → 𝐴 ∈ No )
noseqinds.3	⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))
noseqinds.4	⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))
noseqinds.5	⊢ (𝑦 = (𝑧 +s 1s ) → (𝜓 ↔ 𝜏))
noseqinds.6	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜂))
noseqinds.7	⊢ (𝜑 → 𝜒)
noseqinds.8	⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → (𝜃 → 𝜏))

Assertion

Ref	Expression
noseqinds	⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝜂)

Distinct variable groups:   𝑦,𝑧,𝐴   𝑦,𝐵   𝜒,𝑦   𝜂,𝑦   𝜑,𝑧   𝜓,𝑧   𝜏,𝑦   𝜃,𝑦   𝑦,𝑍,𝑧   𝑥,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑧)   𝜃(𝑥,𝑧)   𝜏(𝑥,𝑧)   𝜂(𝑥,𝑧)   𝐴(𝑥)   𝐵(𝑥,𝑧)   𝑍(𝑥)

Proof of Theorem noseqinds

Step	Hyp	Ref	Expression
1		noseq.1	. . . . 5 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
2		noseq.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ No )
3		noseqinds.3	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))
4	1, 2	noseq0 28213	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑍)
5		noseqinds.7	. . . . . 6 ⊢ (𝜑 → 𝜒)
6	3, 4, 5	elrabd 3681	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ {𝑦 ∈ 𝑍 ∣ 𝜓})
7		noseqinds.8	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → (𝜃 → 𝜏))
8	1	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
9	2	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝐴 ∈ No )
10		simpr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝑧 ∈ 𝑍)
11	8, 9, 10	noseqp1 28214	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → (𝑧 +s 1s ) ∈ 𝑍)
12	7, 11	jctild 524	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → (𝜃 → ((𝑧 +s 1s ) ∈ 𝑍 ∧ 𝜏)))
13	12	expimpd 452	. . . . . . 7 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ∧ 𝜃) → ((𝑧 +s 1s ) ∈ 𝑍 ∧ 𝜏)))
14		noseqinds.4	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))
15	14	elrab 3679	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ 𝑍 ∣ 𝜓} ↔ (𝑧 ∈ 𝑍 ∧ 𝜃))
16		noseqinds.5	. . . . . . . 8 ⊢ (𝑦 = (𝑧 +s 1s ) → (𝜓 ↔ 𝜏))
17	16	elrab 3679	. . . . . . 7 ⊢ ((𝑧 +s 1s ) ∈ {𝑦 ∈ 𝑍 ∣ 𝜓} ↔ ((𝑧 +s 1s ) ∈ 𝑍 ∧ 𝜏))
18	13, 15, 17	3imtr4g 295	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ {𝑦 ∈ 𝑍 ∣ 𝜓} → (𝑧 +s 1s ) ∈ {𝑦 ∈ 𝑍 ∣ 𝜓}))
19	18	imp 405	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑦 ∈ 𝑍 ∣ 𝜓}) → (𝑧 +s 1s ) ∈ {𝑦 ∈ 𝑍 ∣ 𝜓})
20	1, 2, 6, 19	noseqind 28215	. . . 4 ⊢ (𝜑 → 𝑍 ⊆ {𝑦 ∈ 𝑍 ∣ 𝜓})
21	20	sselda 3976	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝐵 ∈ {𝑦 ∈ 𝑍 ∣ 𝜓})
22		noseqinds.6	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜂))
23	22	elrab 3679	. . 3 ⊢ (𝐵 ∈ {𝑦 ∈ 𝑍 ∣ 𝜓} ↔ (𝐵 ∈ 𝑍 ∧ 𝜂))
24	21, 23	sylib 217	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐵 ∈ 𝑍 ∧ 𝜂))
25	24	simprd 494	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝜂)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {crab 3418  Vcvv 3461   ↦ cmpt 5232   “ cima 5681  (class class class)co 7419  ωcom 7871  reccrdg 8430   No csur 27618   1_s c1s 27802   +_s cadds 27922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431

This theorem is referenced by:  n0sind  28254