Theorem rankscottu 35753

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 9949	. . . . . 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 35747	. . . . . 6 ⊢ ((𝑥 ∈ Scott 𝐵 ∧ 𝐴 ∈ 𝐵) → (rank‘𝑥) ⊆ (rank‘𝐴))
5		rankon 9803	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ On
6	5	onordi 6476	. . . . . . 7 ⊢ Ord (rank‘𝑥)
7		rankon 9803	. . . . . . . 8 ⊢ (rank‘𝐴) ∈ On
8	7	onordi 6476	. . . . . . 7 ⊢ Ord (rank‘𝐴)
9		ordsucsssuc 7834	. . . . . . 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 3965	. . . 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 4299	. . . . 5 ⊢ (¬ Scott 𝐵 = ∅ ↔ ∃𝑥 𝑥 ∈ Scott 𝐵)
16	15	con1bii 359	. . . 4 ⊢ (¬ ∃𝑥 𝑥 ∈ Scott 𝐵 ↔ Scott 𝐵 = ∅)
17		scottex 9933	. . . . . 6 ⊢ Scott 𝐵 ∈ V
18	17	rankeq0 9877	. . . . 5 ⊢ (Scott 𝐵 = ∅ ↔ (rank‘Scott 𝐵) = ∅)
19		0ss 4350	. . . . . 6 ⊢ ∅ ⊆ suc (rank‘𝐴)
20		sseq1 3956	. . . . . 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 2145   ⊆ wss 3899  ∅c0 4279  Ord word 6361  suc csuc 6364  ‘cfv 6538  rankcrnk 9767  Scott cscott 9928

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-reg 9586  ax-inf2 9642

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768  df-rank 9769  df-scott 9929

This theorem is used by:  scottssr1  35754  rankkardu  35839