Theorem scottex 9970

Description: Scott's trick produces a set. (Contributed by NM, 13-Oct-2003.) Use the Scott operation. (Revised by BTernaryTau, 18-Jul-2026.)

Assertion

Ref	Expression
scottex	⊢ Scott 𝐴 ∈ V

Proof of Theorem scottex

Dummy variables 𝑣 𝑥 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-scott 9966	. 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
2		0ex 5267	. . . . 5 ⊢ ∅ ∈ V
3		eleq1 2852	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
4	2, 3	mpbiri 261	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
5		rabexg 5306	. . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
6	4, 5	syl 18	. . 3 ⊢ (𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
7		neq0 4302	. . . 4 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑣 𝑣 ∈ 𝐴)
8		fveq2 6916	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (rank‘𝑦) = (rank‘𝑣))
9	8	sseq2d 3945	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑣)))
10	9	rspcv 3576	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑣)))
11	10	adantr 486	. . . . . . 7 ⊢ ((𝑣 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑣)))
12	11	ss2rabdv 4003	. . . . . 6 ⊢ (𝑣 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)})
13		rankon 9840	. . . . . . . . 9 ⊢ (rank‘𝑣) ∈ On
14		fveq2 6916	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (rank‘𝑥) = (rank‘𝑤))
15	14	sseq1d 3944	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((rank‘𝑥) ⊆ (rank‘𝑣) ↔ (rank‘𝑤) ⊆ (rank‘𝑣)))
16	15	elrab 3648	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} ↔ (𝑤 ∈ 𝐴 ∧ (rank‘𝑤) ⊆ (rank‘𝑣)))
17	16	simprbi 503	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} → (rank‘𝑤) ⊆ (rank‘𝑣))
18	17	rgen 3082	. . . . . . . . 9 ⊢ ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} (rank‘𝑤) ⊆ (rank‘𝑣)
19		sseq2 3939	. . . . . . . . . . 11 ⊢ (𝑧 = (rank‘𝑣) → ((rank‘𝑤) ⊆ 𝑧 ↔ (rank‘𝑤) ⊆ (rank‘𝑣)))
20	19	ralbidv 3189	. . . . . . . . . 10 ⊢ (𝑧 = (rank‘𝑣) → (∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} (rank‘𝑤) ⊆ 𝑧 ↔ ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} (rank‘𝑤) ⊆ (rank‘𝑣)))
21	20	rspcev 3580	. . . . . . . . 9 ⊢ (((rank‘𝑣) ∈ On ∧ ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} (rank‘𝑤) ⊆ (rank‘𝑣)) → ∃𝑧 ∈ On ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} (rank‘𝑤) ⊆ 𝑧)
22	13, 18, 21	mp2an 705	. . . . . . . 8 ⊢ ∃𝑧 ∈ On ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} (rank‘𝑤) ⊆ 𝑧
23		bndrank 9891	. . . . . . . 8 ⊢ (∃𝑧 ∈ On ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} (rank‘𝑤) ⊆ 𝑧 → {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} ∈ V)
24	22, 23	ax-mp 5	. . . . . . 7 ⊢ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} ∈ V
25	24	ssex 5289	. . . . . 6 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
26	12, 25	syl 18	. . . . 5 ⊢ (𝑣 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
27	26	exlimiv 1965	. . . 4 ⊢ (∃𝑣 𝑣 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
28	7, 27	sylbi 220	. . 3 ⊢ (¬ 𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
29	6, 28	pm2.61i 184	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
30	1, 29	eqeltri 2860	1 ⊢ Scott 𝐴 ∈ V

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  Oncon0 6388  ‘cfv 6566  rankcrnk 9804  Scott cscott 9965

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-reg 9623  ax-inf2 9679

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 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-r1 9805  df-rank 9806  df-scott 9966

This theorem is used by:  scottrankd  9986  cplem2  9989  kardex  9994  hta  9999  rankscottu  35880  scottssr1  35881  kardfn  35960  kardval  35961  scottexf  39366  gruscottcld  45494  collexd  45502