Theorem scottelrankd 44216

Description: Property of a Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)

Hypotheses

Ref	Expression
scottelrankd.1	⊢ (𝜑 → 𝐵 ∈ Scott 𝐴)
scottelrankd.2	⊢ (𝜑 → 𝐶 ∈ Scott 𝐴)

Assertion

Ref	Expression
scottelrankd	⊢ (𝜑 → (rank‘𝐵) ⊆ (rank‘𝐶))

Proof of Theorem scottelrankd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6920	. . 3 ⊢ (𝑦 = 𝐶 → (rank‘𝑦) = (rank‘𝐶))
2	1	sseq2d 4041	. 2 ⊢ (𝑦 = 𝐶 → ((rank‘𝐵) ⊆ (rank‘𝑦) ↔ (rank‘𝐵) ⊆ (rank‘𝐶)))
3		scottelrankd.1	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴)
4		df-scott 44205	. . . . 5 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
5	3, 4	eleqtrdi 2854	. . . 4 ⊢ (𝜑 → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
6		fveq2 6920	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (rank‘𝑥) = (rank‘𝐵))
7	6	sseq1d 4040	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝐵) ⊆ (rank‘𝑦)))
8	7	ralbidv 3184	. . . . 5 ⊢ (𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (rank‘𝐵) ⊆ (rank‘𝑦)))
9	8	elrab 3708	. . . 4 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (rank‘𝐵) ⊆ (rank‘𝑦)))
10	5, 9	sylib 218	. . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (rank‘𝐵) ⊆ (rank‘𝑦)))
11	10	simprd 495	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (rank‘𝐵) ⊆ (rank‘𝑦))
12		scottelrankd.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Scott 𝐴)
13	12, 4	eleqtrdi 2854	. . 3 ⊢ (𝜑 → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
14		elrabi 3703	. . 3 ⊢ (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} → 𝐶 ∈ 𝐴)
15	13, 14	syl 17	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
16	2, 11, 15	rspcdva 3636	1 ⊢ (𝜑 → (rank‘𝐵) ⊆ (rank‘𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443   ⊆ wss 3976  ‘cfv 6573  rankcrnk 9832  Scott cscott 44204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-scott 44205

This theorem is referenced by:  scottrankd  44217