Theorem dfscott3 35521

Description: Alternate definition of a Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)

Assertion

Ref	Expression
dfscott3	⊢ Scott 𝐴 = (𝐴 ∩ (𝑅1‘suc ∩ (rank “ 𝐴)))

Proof of Theorem dfscott3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfscott2 35520	. 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) = ∩ (rank “ 𝐴)}
2		rankfn 35515	. . . . . . . . . . 11 ⊢ rank Fn V
3		ssv 3960	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ V
4		fnfvima 7231	. . . . . . . . . . 11 ⊢ ((rank Fn V ∧ 𝐴 ⊆ V ∧ 𝑥 ∈ 𝐴) → (rank‘𝑥) ∈ (rank “ 𝐴))
5	2, 3, 4	mp3an12 1479	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank “ 𝐴))
6		intss1 4927	. . . . . . . . . 10 ⊢ ((rank‘𝑥) ∈ (rank “ 𝐴) → ∩ (rank “ 𝐴) ⊆ (rank‘𝑥))
7	5, 6	syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ∩ (rank “ 𝐴) ⊆ (rank‘𝑥))
8		ne0i 4293	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
9		rankfo 35514	. . . . . . . . . . . . . . . . 17 ⊢ rank:V–onto→On
10		fof 6792	. . . . . . . . . . . . . . . . 17 ⊢ (rank:V–onto→On → rank:V⟶On)
11	9, 10	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ rank:V⟶On
12	11	fdmi 6717	. . . . . . . . . . . . . . 15 ⊢ dom rank = V
13	12	ineq1i 4168	. . . . . . . . . . . . . 14 ⊢ (dom rank ∩ 𝐴) = (V ∩ 𝐴)
14		inv2 35476	. . . . . . . . . . . . . 14 ⊢ (V ∩ 𝐴) = 𝐴
15	13, 14	eqtri 2785	. . . . . . . . . . . . 13 ⊢ (dom rank ∩ 𝐴) = 𝐴
16	15	neeq1i 3021	. . . . . . . . . . . 12 ⊢ ((dom rank ∩ 𝐴) ≠ ∅ ↔ 𝐴 ≠ ∅)
17	16	biimpri 231	. . . . . . . . . . 11 ⊢ (𝐴 ≠ ∅ → (dom rank ∩ 𝐴) ≠ ∅)
18	17	imadisjlnd 6082	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ → (rank “ 𝐴) ≠ ∅)
19		fimass 6726	. . . . . . . . . . . 12 ⊢ (rank:V⟶On → (rank “ 𝐴) ⊆ On)
20	11, 19	ax-mp 5	. . . . . . . . . . 11 ⊢ (rank “ 𝐴) ⊆ On
21		oninton 7792	. . . . . . . . . . 11 ⊢ (((rank “ 𝐴) ⊆ On ∧ (rank “ 𝐴) ≠ ∅) → ∩ (rank “ 𝐴) ∈ On)
22	20, 21	mpan 702	. . . . . . . . . 10 ⊢ ((rank “ 𝐴) ≠ ∅ → ∩ (rank “ 𝐴) ∈ On)
23		vex 3458	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
24	23	ssrankr1 9805	. . . . . . . . . 10 ⊢ (∩ (rank “ 𝐴) ∈ On → (∩ (rank “ 𝐴) ⊆ (rank‘𝑥) ↔ ¬ 𝑥 ∈ (𝑅1‘∩ (rank “ 𝐴))))
25	8, 18, 22, 24	4syl 20	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (∩ (rank “ 𝐴) ⊆ (rank‘𝑥) ↔ ¬ 𝑥 ∈ (𝑅1‘∩ (rank “ 𝐴))))
26	7, 25	mpbid 235	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ (𝑅1‘∩ (rank “ 𝐴)))
27	26	biantrurd 541	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ (𝑅1‘suc ∩ (rank “ 𝐴)) ↔ (¬ 𝑥 ∈ (𝑅1‘∩ (rank “ 𝐴)) ∧ 𝑥 ∈ (𝑅1‘suc ∩ (rank “ 𝐴)))))
28	23	rankr1 9804	. . . . . . 7 ⊢ (∩ (rank “ 𝐴) = (rank‘𝑥) ↔ (¬ 𝑥 ∈ (𝑅1‘∩ (rank “ 𝐴)) ∧ 𝑥 ∈ (𝑅1‘suc ∩ (rank “ 𝐴))))
29	27, 28	bitr4di 292	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ (𝑅1‘suc ∩ (rank “ 𝐴)) ↔ ∩ (rank “ 𝐴) = (rank‘𝑥)))
30		eqcom 2769	. . . . . 6 ⊢ ((rank‘𝑥) = ∩ (rank “ 𝐴) ↔ ∩ (rank “ 𝐴) = (rank‘𝑥))
31	29, 30	bitr4di 292	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ (𝑅1‘suc ∩ (rank “ 𝐴)) ↔ (rank‘𝑥) = ∩ (rank “ 𝐴)))
32	31	adantl 486	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝑅1‘suc ∩ (rank “ 𝐴)) ↔ (rank‘𝑥) = ∩ (rank “ 𝐴)))
33	32	rabbi2dva 4177	. . 3 ⊢ (⊤ → (𝐴 ∩ (𝑅1‘suc ∩ (rank “ 𝐴))) = {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) = ∩ (rank “ 𝐴)})
34	33	mptru 1576	. 2 ⊢ (𝐴 ∩ (𝑅1‘suc ∩ (rank “ 𝐴))) = {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) = ∩ (rank “ 𝐴)}
35	1, 34	eqtr4i 2788	1 ⊢ Scott 𝐴 = (𝐴 ∩ (𝑅1‘suc ∩ (rank “ 𝐴)))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 400   = wceq 1569  ⊤wtru 1570   ∈ wcel 2142   ≠ wne 2957  {crab 3415  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ∩ cint 4911  dom cdm 5660   “ cima 5663  Oncon0 6360  suc csuc 6362   Fn wfn 6531  ⟶wf 6532  –onto→wfo 6534  ‘cfv 6536  𝑅₁cr1 9732  rankcrnk 9733  Scott cscott 9855

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-reg 9552  ax-inf2 9608

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-r1 9734  df-rank 9735  df-scott 9856

This theorem is used by: (None)