Theorem dfscott3 35613

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 35612	. 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) = ∩ (rank “ 𝐴)}
2		rankfn 35607	. . . . . . . . . . 11 ⊢ rank Fn V
3		ssv 3958	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ V
4		fnfvima 7235	. . . . . . . . . . 11 ⊢ ((rank Fn V ∧ 𝐴 ⊆ V ∧ 𝑥 ∈ 𝐴) → (rank‘𝑥) ∈ (rank “ 𝐴))
5	2, 3, 4	mp3an12 1480	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank “ 𝐴))
6		intss1 4926	. . . . . . . . . 10 ⊢ ((rank‘𝑥) ∈ (rank “ 𝐴) → ∩ (rank “ 𝐴) ⊆ (rank‘𝑥))
7	5, 6	syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ∩ (rank “ 𝐴) ⊆ (rank‘𝑥))
8		ne0i 4290	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
9		rankfo 35606	. . . . . . . . . . . . . . . . 17 ⊢ rank:V–onto→On
10		fof 6793	. . . . . . . . . . . . . . . . 17 ⊢ (rank:V–onto→On → rank:V⟶On)
11	9, 10	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ rank:V⟶On
12	11	fdmi 6718	. . . . . . . . . . . . . . 15 ⊢ dom rank = V
13	12	ineq1i 4165	. . . . . . . . . . . . . 14 ⊢ (dom rank ∩ 𝐴) = (V ∩ 𝐴)
14		inv2 35575	. . . . . . . . . . . . . 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 6081	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ → (rank “ 𝐴) ≠ ∅)
19		fimass 6727	. . . . . . . . . . . 12 ⊢ (rank:V⟶On → (rank “ 𝐴) ⊆ On)
20	11, 19	ax-mp 5	. . . . . . . . . . 11 ⊢ (rank “ 𝐴) ⊆ On
21		oninton 7797	. . . . . . . . . . 11 ⊢ (((rank “ 𝐴) ⊆ On ∧ (rank “ 𝐴) ≠ ∅) → ∩ (rank “ 𝐴) ∈ On)
22	20, 21	mpan 703	. . . . . . . . . 10 ⊢ ((rank “ 𝐴) ≠ ∅ → ∩ (rank “ 𝐴) ∈ On)
23		vex 3457	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
24	23	ssrankr1 9820	. . . . . . . . . 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 542	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ (𝑅1‘suc ∩ (rank “ 𝐴)) ↔ (¬ 𝑥 ∈ (𝑅1‘∩ (rank “ 𝐴)) ∧ 𝑥 ∈ (𝑅1‘suc ∩ (rank “ 𝐴)))))
28	23	rankr1 9819	. . . . . . 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 487	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝑅1‘suc ∩ (rank “ 𝐴)) ↔ (rank‘𝑥) = ∩ (rank “ 𝐴)))
33	32	rabbi2dva 4174	. . 3 ⊢ (⊤ → (𝐴 ∩ (𝑅1‘suc ∩ (rank “ 𝐴))) = {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) = ∩ (rank “ 𝐴)})
34	33	mptru 1577	. 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 401   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145   ≠ wne 2957  {crab 3414  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  ∩ cint 4910  dom cdm 5659   “ cima 5662  Oncon0 6361  suc csuc 6363   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537  𝑅₁cr1 9747  rankcrnk 9748  Scott cscott 9870

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-reg 9567  ax-inf2 9623

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 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 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-r1 9749  df-rank 9750  df-scott 9871

This theorem is used by: (None)