Theorem rankscottu 35456

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 9874	. . . . . 6 ⊢ (𝑥 ∈ Scott 𝐵 → (rank‘Scott 𝐵) = suc (rank‘𝑥))
3	2	adantr 485	. . . . 5 ⊢ ((𝑥 ∈ Scott 𝐵 ∧ 𝐴 ∈ 𝐵) → (rank‘Scott 𝐵) = suc (rank‘𝑥))
4		elscottrankss 35452	. . . . . 6 ⊢ ((𝑥 ∈ Scott 𝐵 ∧ 𝐴 ∈ 𝐵) → (rank‘𝑥) ⊆ (rank‘𝐴))
5		rankon 9767	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ On
6	5	onordi 6475	. . . . . . 7 ⊢ Ord (rank‘𝑥)
7		rankon 9767	. . . . . . . 8 ⊢ (rank‘𝐴) ∈ On
8	7	onordi 6475	. . . . . . 7 ⊢ Ord (rank‘𝐴)
9		ordsucsssuc 7819	. . . . . . 7 ⊢ ((Ord (rank‘𝑥) ∧ Ord (rank‘𝐴)) → ((rank‘𝑥) ⊆ (rank‘𝐴) ↔ suc (rank‘𝑥) ⊆ suc (rank‘𝐴)))
10	6, 8, 9	mp2an 704	. . . . . 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 3979	. . . 4 ⊢ ((𝑥 ∈ Scott 𝐵 ∧ 𝐴 ∈ 𝐵) → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴))
13	12	ex 417	. . 3 ⊢ (𝑥 ∈ Scott 𝐵 → (𝐴 ∈ 𝐵 → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴)))
14	13	exlimiv 1957	. 2 ⊢ (∃𝑥 𝑥 ∈ Scott 𝐵 → (𝐴 ∈ 𝐵 → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴)))
15		neq0 4314	. . . . 5 ⊢ (¬ Scott 𝐵 = ∅ ↔ ∃𝑥 𝑥 ∈ Scott 𝐵)
16	15	con1bii 359	. . . 4 ⊢ (¬ ∃𝑥 𝑥 ∈ Scott 𝐵 ↔ Scott 𝐵 = ∅)
17		scottex2 9871	. . . . . 6 ⊢ Scott 𝐵 ∈ V
18	17	rankeq0 9833	. . . . 5 ⊢ (Scott 𝐵 = ∅ ↔ (rank‘Scott 𝐵) = ∅)
19		0ss 4364	. . . . . 6 ⊢ ∅ ⊆ suc (rank‘𝐴)
20		sseq1 3970	. . . . . 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))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ⊆ wss 3913  ∅c0 4294  Ord word 6360  suc csuc 6363  ‘cfv 6537  rankcrnk 9735  Scott cscott 9857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-reg 9554  ax-inf2 9610

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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 7414  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-r1 9736  df-rank 9737  df-scott 9858

This theorem is referenced by:  scottssr1  35457  rankkardu  35517