Theorem scottex 9865

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 9861	. 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
2		0ex 5272	. . . . 5 ⊢ ∅ ∈ V
3		eleq1 2853	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
4	2, 3	mpbiri 261	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
5		rabexg 5310	. . . 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 4306	. . . 4 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑣 𝑣 ∈ 𝐴)
8		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (rank‘𝑦) = (rank‘𝑣))
9	8	sseq2d 3970	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑣)))
10	9	rspcv 3579	. . . . . . . 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 4030	. . . . . 6 ⊢ (𝑣 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)})
13		rankon 9770	. . . . . . . . 9 ⊢ (rank‘𝑣) ∈ On
14		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (rank‘𝑥) = (rank‘𝑤))
15	14	sseq1d 3969	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((rank‘𝑥) ⊆ (rank‘𝑣) ↔ (rank‘𝑤) ⊆ (rank‘𝑣)))
16	15	elrab 3652	. . . . . . . . . . 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 3083	. . . . . . . . 9 ⊢ ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} (rank‘𝑤) ⊆ (rank‘𝑣)
19		sseq2 3964	. . . . . . . . . . 11 ⊢ (𝑧 = (rank‘𝑣) → ((rank‘𝑤) ⊆ 𝑧 ↔ (rank‘𝑤) ⊆ (rank‘𝑣)))
20	19	ralbidv 3190	. . . . . . . . . 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 3583	. . . . . . . . 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 9816	. . . . . . . 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 5293	. . . . . 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 2861	1 ⊢ Scott 𝐴 ∈ V

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  Oncon0 6364  ‘cfv 6540  rankcrnk 9738  Scott cscott 9860

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-reg 9557  ax-inf2 9613

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-om 7865  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-r1 9739  df-rank 9740  df-scott 9861

This theorem is used by:  scottrankd  9881  cplem2  9884  kardex  9889  hta  9894  rankscottu  35539  scottssr1  35540  kardfn  35580  kardval  35581  scottexf  38850  gruscottcld  44992  collexd  45000