Theorem noseqex 28705

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 8452	. . 3 ⊢ Fun rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴)
3		dcomex 10552	. . . 4 ⊢ ω ∈ V
4	3	funimaex 6649	. . 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 2872	1 ⊢ (𝜑 → 𝑍 ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ↦ cmpt 5189   “ cima 5658  Fun wfun 6552  (class class class)co 7443  ωcom 7901  reccrdg 8445   1_s c1s 28222   +_s cadds 28375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775  ax-dc 10551

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504

This theorem is used by:  om2noseqoi  28719