Theorem noseqex 28493

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 8401	. . 3 ⊢ Fun rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴)
3		dcomex 10437	. . . 4 ⊢ ω ∈ V
4	3	funimaex 6623	. . 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 2870	1 ⊢ (𝜑 → 𝑍 ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  Vcvv 3454   ↦ cmpt 5191   “ cima 5663  Fun wfun 6530  (class class class)co 7412  ωcom 7860  reccrdg 8394   1_s c1s 28010   +_s cadds 28163

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734  ax-dc 10436

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451

This theorem is used by:  om2noseqoi  28507