Theorem scottex 9926

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 9922	. 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
2		0ex 5261	. . . . 5 ⊢ ∅ ∈ V
3		eleq1 2849	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
4	2, 3	mpbiri 261	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
5		rabexg 5299	. . . 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 4299	. . . 4 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑣 𝑣 ∈ 𝐴)
8		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (rank‘𝑦) = (rank‘𝑣))
9	8	sseq2d 3963	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑣)))
10	9	rspcv 3573	. . . . . . . 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 4023	. . . . . 6 ⊢ (𝑣 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)})
13		rankon 9796	. . . . . . . . 9 ⊢ (rank‘𝑣) ∈ On
14		fveq2 6883	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (rank‘𝑥) = (rank‘𝑤))
15	14	sseq1d 3962	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((rank‘𝑥) ⊆ (rank‘𝑣) ↔ (rank‘𝑤) ⊆ (rank‘𝑣)))
16	15	elrab 3645	. . . . . . . . . . 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 3079	. . . . . . . . 9 ⊢ ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} (rank‘𝑤) ⊆ (rank‘𝑣)
19		sseq2 3957	. . . . . . . . . . 11 ⊢ (𝑧 = (rank‘𝑣) → ((rank‘𝑤) ⊆ 𝑧 ↔ (rank‘𝑤) ⊆ (rank‘𝑣)))
20	19	ralbidv 3186	. . . . . . . . . 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 3577	. . . . . . . . 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 9847	. . . . . . . 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 5282	. . . . . 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 1963	. . . 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 2857	1 ⊢ Scott 𝐴 ∈ V

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  Oncon0 6361  ‘cfv 6537  rankcrnk 9760  Scott cscott 9921

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-reg 9579  ax-inf2 9635

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-r1 9761  df-rank 9762  df-scott 9922

This theorem is used by:  scottrankd  9942  cplem2  9945  kardex  9950  hta  9955  rankscottu  35741  scottssr1  35742  kardfn  35802  kardval  35803  scottexf  39080  gruscottcld  45218  collexd  45226