Theorem rankscottu 35523

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 9870	. . . . . 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 35516	. . . . . 6 ⊢ ((𝑥 ∈ Scott 𝐵 ∧ 𝐴 ∈ 𝐵) → (rank‘𝑥) ⊆ (rank‘𝐴))
5		rankon 9763	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ On
6	5	onordi 6474	. . . . . . 7 ⊢ Ord (rank‘𝑥)
7		rankon 9763	. . . . . . . 8 ⊢ (rank‘𝐴) ∈ On
8	7	onordi 6474	. . . . . . 7 ⊢ Ord (rank‘𝐴)
9		ordsucsssuc 7815	. . . . . . 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 3971	. . . 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 1960	. 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		scottex2 9867	. . . . . 6 ⊢ Scott 𝐵 ∈ V
18	17	rankeq0 9829	. . . . 5 ⊢ (Scott 𝐵 = ∅ ↔ (rank‘Scott 𝐵) = ∅)
19		0ss 4357	. . . . . 6 ⊢ ∅ ⊆ suc (rank‘𝐴)
20		sseq1 3962	. . . . . 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 1570  ∃wex 1809   ∈ wcel 2143   ⊆ wss 3905  ∅c0 4286  Ord word 6359  suc csuc 6362  ‘cfv 6536  rankcrnk 9731  Scott cscott 9853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-reg 9550  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  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 7413  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-r1 9732  df-rank 9733  df-scott 9854

This theorem is referenced by:  scottssr1  35524  rankkardu  35584