Theorem noseqex 28183

Description: The next several theorems develop the concept of a countable sequence of surreals. This set is denoted by 𝑍 here and is the analogue of the upper integer sets for surreal numbers. However, we do not require the starting point to be an integer so we can accommodate infinite numbers. This first theorem establishes that 𝑍 is a set. (Contributed by Scott Fenton, 18-Apr-2025.)

Hypothesis

Ref	Expression
noseq.1	⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))

Assertion

Ref	Expression
noseqex	⊢ (𝜑 → 𝑍 ∈ V)

Proof of Theorem noseqex

Step	Hyp	Ref	Expression
1		noseq.1	. 2 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
2		rdgfun 8384	. . 3 ⊢ Fun rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴)
3		dcomex 10400	. . . 4 ⊢ ω ∈ V
4	3	funimaex 6605	. . 3 ⊢ (Fun rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) → (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω) ∈ V)
5	2, 4	ax-mp 5	. 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω) ∈ V
6	1, 5	eqeltrdi 2836	1 ⊢ (𝜑 → 𝑍 ∈ V)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ↦ cmpt 5188   “ cima 5641  Fun wfun 6505  (class class class)co 7387  ωcom 7842  reccrdg 8377   1_s c1s 27735   +_s cadds 27866

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711  ax-dc 10399

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434

This theorem is referenced by:  om2noseqoi  28197  n0sex  28210