Theorem rankscottu 35582

Description: An upper bound on the rank of a Scott's trick set. (Contributed by BTernaryTau, 4-Jul-2026.)

Assertion

Ref	Expression
rankscottu	⊢ (𝐴 ∈ 𝐵 → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴))

Proof of Theorem rankscottu

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. . . . . . 7 ⊢ (𝑥 ∈ Scott 𝐵 → 𝑥 ∈ Scott 𝐵)
2	1	scottrankd 9885	. . . . . 6 ⊢ (𝑥 ∈ Scott 𝐵 → (rank‘Scott 𝐵) = suc (rank‘𝑥))
3	2	adantr 486	. . . . 5 ⊢ ((𝑥 ∈ Scott 𝐵 ∧ 𝐴 ∈ 𝐵) → (rank‘Scott 𝐵) = suc (rank‘𝑥))
4		elscottrankss 35576	. . . . . 6 ⊢ ((𝑥 ∈ Scott 𝐵 ∧ 𝐴 ∈ 𝐵) → (rank‘𝑥) ⊆ (rank‘𝐴))
5		rankon 9774	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ On
6	5	onordi 6478	. . . . . . 7 ⊢ Ord (rank‘𝑥)
7		rankon 9774	. . . . . . . 8 ⊢ (rank‘𝐴) ∈ On
8	7	onordi 6478	. . . . . . 7 ⊢ Ord (rank‘𝐴)
9		ordsucsssuc 7825	. . . . . . 7 ⊢ ((Ord (rank‘𝑥) ∧ Ord (rank‘𝐴)) → ((rank‘𝑥) ⊆ (rank‘𝐴) ↔ suc (rank‘𝑥) ⊆ suc (rank‘𝐴)))
10	6, 8, 9	mp2an 705	. . . . . 6 ⊢ ((rank‘𝑥) ⊆ (rank‘𝐴) ↔ suc (rank‘𝑥) ⊆ suc (rank‘𝐴))
11	4, 10	sylib 221	. . . . 5 ⊢ ((𝑥 ∈ Scott 𝐵 ∧ 𝐴 ∈ 𝐵) → suc (rank‘𝑥) ⊆ suc (rank‘𝐴))
12	3, 11	eqsstrd 3972	. . . 4 ⊢ ((𝑥 ∈ Scott 𝐵 ∧ 𝐴 ∈ 𝐵) → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴))
13	12	ex 418	. . 3 ⊢ (𝑥 ∈ Scott 𝐵 → (𝐴 ∈ 𝐵 → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴)))
14	13	exlimiv 1963	. 2 ⊢ (∃𝑥 𝑥 ∈ Scott 𝐵 → (𝐴 ∈ 𝐵 → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴)))
15		neq0 4306	. . . . 5 ⊢ (¬ Scott 𝐵 = ∅ ↔ ∃𝑥 𝑥 ∈ Scott 𝐵)
16	15	con1bii 359	. . . 4 ⊢ (¬ ∃𝑥 𝑥 ∈ Scott 𝐵 ↔ Scott 𝐵 = ∅)
17		scottex 9869	. . . . . 6 ⊢ Scott 𝐵 ∈ V
18	17	rankeq0 9840	. . . . 5 ⊢ (Scott 𝐵 = ∅ ↔ (rank‘Scott 𝐵) = ∅)
19		0ss 4357	. . . . . 6 ⊢ ∅ ⊆ suc (rank‘𝐴)
20		sseq1 3963	. . . . . 6 ⊢ ((rank‘Scott 𝐵) = ∅ → ((rank‘Scott 𝐵) ⊆ suc (rank‘𝐴) ↔ ∅ ⊆ suc (rank‘𝐴)))
21	19, 20	mpbiri 261	. . . . 5 ⊢ ((rank‘Scott 𝐵) = ∅ → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴))
22	18, 21	sylbi 220	. . . 4 ⊢ (Scott 𝐵 = ∅ → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴))
23	16, 22	sylbi 220	. . 3 ⊢ (¬ ∃𝑥 𝑥 ∈ Scott 𝐵 → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴))
24	23	a1d 26	. 2 ⊢ (¬ ∃𝑥 𝑥 ∈ Scott 𝐵 → (𝐴 ∈ 𝐵 → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴)))
25	14, 24	pm2.61i 184	1 ⊢ (𝐴 ∈ 𝐵 → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ⊆ wss 3906  ∅c0 4286  Ord word 6363  suc csuc 6366  ‘cfv 6540  rankcrnk 9742  Scott cscott 9864

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 7742  ax-reg 9561  ax-inf2 9617

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 7422  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-r1 9743  df-rank 9744  df-scott 9865

This theorem is used by:  scottssr1  35583  rankkardu  35643