Theorem dfscott3 35673

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 35672	. 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) = ∩ (rank “ 𝐴)}
2		rankfn 35667	. . . . . . . . . . 11 ⊢ rank Fn V
3		ssv 3954	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ V
4		fnfvima 7227	. . . . . . . . . . 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 4922	. . . . . . . . . 10 ⊢ ((rank‘𝑥) ∈ (rank “ 𝐴) → ∩ (rank “ 𝐴) ⊆ (rank‘𝑥))
7	5, 6	syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ∩ (rank “ 𝐴) ⊆ (rank‘𝑥))
8		ne0i 4286	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
9		rankfo 35666	. . . . . . . . . . . . . . . . 17 ⊢ rank:V–onto→On
10		fof 6784	. . . . . . . . . . . . . . . . 17 ⊢ (rank:V–onto→On → rank:V⟶On)
11	9, 10	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ rank:V⟶On
12	11	fdmi 6709	. . . . . . . . . . . . . . 15 ⊢ dom rank = V
13	12	ineq1i 4161	. . . . . . . . . . . . . 14 ⊢ (dom rank ∩ 𝐴) = (V ∩ 𝐴)
14		inv2 35643	. . . . . . . . . . . . . 14 ⊢ (V ∩ 𝐴) = 𝐴
15	13, 14	eqtri 2783	. . . . . . . . . . . . 13 ⊢ (dom rank ∩ 𝐴) = 𝐴
16	15	neeq1i 3019	. . . . . . . . . . . 12 ⊢ ((dom rank ∩ 𝐴) ≠ ∅ ↔ 𝐴 ≠ ∅)
17	16	biimpri 231	. . . . . . . . . . 11 ⊢ (𝐴 ≠ ∅ → (dom rank ∩ 𝐴) ≠ ∅)
18	17	imadisjlnd 6071	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ → (rank “ 𝐴) ≠ ∅)
19		fimass 6718	. . . . . . . . . . . 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 703	. . . . . . . . . 10 ⊢ ((rank “ 𝐴) ≠ ∅ → ∩ (rank “ 𝐴) ∈ On)
23		vex 3454	. . . . . . . . . . 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 2767	. . . . . 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 4170	. . 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 2786	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 2955  {crab 3412  Vcvv 3450   ∩ cin 3897   ⊆ wss 3898  ∅c0 4278  ∩ cint 4906  dom cdm 5647   “ cima 5650  Oncon0 6351  suc csuc 6353   Fn wfn 6522  ⟶wf 6523  –onto→wfo 6525  ‘cfv 6527  𝑅₁cr1 9744  rankcrnk 9745  Scott cscott 9899

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 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-reg 9564  ax-inf2 9620

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 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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-r1 9746  df-rank 9747  df-scott 9900

This theorem is used by: (None)