Theorem noseqex 28384

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 8389	. . 3 ⊢ Fun rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴)
3		dcomex 10406	. . . 4 ⊢ ω ∈ V
4	3	funimaex 6611	. . 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)

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144  Vcvv 3456   ↦ cmpt 5183   “ cima 5652  Fun wfun 6517  (class class class)co 7398  ωcom 7848  reccrdg 8382   1_s c1s 27901   +_s cadds 28054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720  ax-dc 10405

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439

This theorem is referenced by:  om2noseqoi  28398