Theorem noseqex 28212

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 8437	. . 3 ⊢ Fun rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴)
3		dcomex 10472	. . . 4 ⊢ ω ∈ V
4	3	funimaex 6642	. . 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 2833	1 ⊢ (𝜑 → 𝑍 ∈ V)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3461   ↦ cmpt 5232   “ cima 5681  Fun wfun 6543  (class class class)co 7419  ωcom 7871  reccrdg 8430   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-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741  ax-dc 10471

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-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487

This theorem is referenced by:  om2noseqoi  28226  n0sex  28239