Theorem noseq0 28240

Description: The surreal 𝐴 is a member of the sequence starting at 𝐴. (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 )

Assertion

Ref	Expression
noseq0	⊢ (𝜑 → 𝐴 ∈ 𝑍)

Proof of Theorem noseq0

Step	Hyp	Ref	Expression
1		noseq.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ No )
2		fr0g 8364	. . . 4 ⊢ (𝐴 ∈ No → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) = 𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) = 𝐴)
4		frfnom 8363	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω
5		peano1 7828	. . . 4 ⊢ ∅ ∈ ω
6		fnfvelrn 7022	. . . 4 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
7	4, 5, 6	mp2an 692	. . 3 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
8	3, 7	eqeltrrdi 2842	. 2 ⊢ (𝜑 → 𝐴 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
9		noseq.1	. . 3 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
10		df-ima 5634	. . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
11	9, 10	eqtrdi 2784	. 2 ⊢ (𝜑 → 𝑍 = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
12	8, 11	eleqtrrd 2836	1 ⊢ (𝜑 → 𝐴 ∈ 𝑍)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ∅c0 4282   ↦ cmpt 5176  ran crn 5622   ↾ cres 5623   “ cima 5624   Fn wfn 6484  ‘cfv 6489  (class class class)co 7355  ωcom 7805  reccrdg 8337   No csur 27598   1_s c1s 27787   +_s cadds 27922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-om 7806  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338

This theorem is referenced by:  noseqinds  28243  0n0s  28278