Theorem elscott 35510

Description: Membership in a Scott's trick set. (Contributed by BTernaryTau, 3-Jul-2026.)

Assertion

Ref	Expression
elscott	⊢ (𝐴 ∈ Scott 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (rank‘𝐴) ⊆ (rank‘𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elscott

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . 4 ⊢ (𝑦 = 𝐴 → (rank‘𝑦) = (rank‘𝐴))
2	1	sseq1d 3968	. . 3 ⊢ (𝑦 = 𝐴 → ((rank‘𝑦) ⊆ (rank‘𝑥) ↔ (rank‘𝐴) ⊆ (rank‘𝑥)))
3	2	ralbidv 3188	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑥) ↔ ∀𝑥 ∈ 𝐵 (rank‘𝐴) ⊆ (rank‘𝑥)))
4		df-scott 9854	. 2 ⊢ Scott 𝐵 = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑥)}
5	3, 4	elrab2 3654	1 ⊢ (𝐴 ∈ Scott 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (rank‘𝐴) ⊆ (rank‘𝑥)))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3905  ‘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-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-scott 9854

This theorem is referenced by:  elscottrankss  35516