Theorem scottelrankd 44208

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 6865	. . 3 ⊢ (𝑦 = 𝐶 → (rank‘𝑦) = (rank‘𝐶))
2	1	sseq2d 3987	. 2 ⊢ (𝑦 = 𝐶 → ((rank‘𝐵) ⊆ (rank‘𝑦) ↔ (rank‘𝐵) ⊆ (rank‘𝐶)))
3		scottelrankd.1	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴)
4		df-scott 44197	. . . . 5 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
5	3, 4	eleqtrdi 2839	. . . 4 ⊢ (𝜑 → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
6		fveq2 6865	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (rank‘𝑥) = (rank‘𝐵))
7	6	sseq1d 3986	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝐵) ⊆ (rank‘𝑦)))
8	7	ralbidv 3158	. . . . 5 ⊢ (𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (rank‘𝐵) ⊆ (rank‘𝑦)))
9	8	elrab 3667	. . . 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 2839	. . 3 ⊢ (𝜑 → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
14		elrabi 3662	. . 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 3598	1 ⊢ (𝜑 → (rank‘𝐵) ⊆ (rank‘𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3046  {crab 3411   ⊆ wss 3922  ‘cfv 6519  rankcrnk 9734  Scott cscott 44196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3047  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-iota 6472  df-fv 6527  df-scott 44197

This theorem is referenced by:  scottrankd  44209