Theorem scottex 9937

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 9933	. 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
2		0ex 5260	. . . . 5 ⊢ ∅ ∈ V
3		eleq1 2848	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
4	2, 3	mpbiri 261	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
5		rabexg 5298	. . . 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 4298	. . . 4 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑣 𝑣 ∈ 𝐴)
8		fveq2 6886	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (rank‘𝑦) = (rank‘𝑣))
9	8	sseq2d 3940	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑣)))
10	9	rspcv 3572	. . . . . . . 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 4000	. . . . . 6 ⊢ (𝑣 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)})
13		rankon 9807	. . . . . . . . 9 ⊢ (rank‘𝑣) ∈ On
14		fveq2 6886	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (rank‘𝑥) = (rank‘𝑤))
15	14	sseq1d 3939	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((rank‘𝑥) ⊆ (rank‘𝑣) ↔ (rank‘𝑤) ⊆ (rank‘𝑣)))
16	15	elrab 3644	. . . . . . . . . . 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 3078	. . . . . . . . 9 ⊢ ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑣)} (rank‘𝑤) ⊆ (rank‘𝑣)
19		sseq2 3934	. . . . . . . . . . 11 ⊢ (𝑧 = (rank‘𝑣) → ((rank‘𝑤) ⊆ 𝑧 ↔ (rank‘𝑤) ⊆ (rank‘𝑣)))
20	19	ralbidv 3185	. . . . . . . . . 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 3576	. . . . . . . . 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 9858	. . . . . . . 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 5281	. . . . . 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 1962	. . . 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 2856	1 ⊢ Scott 𝐴 ∈ V

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  Oncon0 6362  ‘cfv 6539  rankcrnk 9771  Scott cscott 9932

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-reg 9590  ax-inf2 9646

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-om 7879  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-r1 9772  df-rank 9773  df-scott 9933

This theorem is used by:  scottrankd  9953  cplem2  9956  kardex  9961  hta  9966  rankscottu  35808  scottssr1  35809  kardfn  35876  kardval  35877  scottexf  39194  gruscottcld  45327  collexd  45335