Theorem elopabran 5581

Description: Membership in an ordered-pair class abstraction defined by a restricted binary relation. (Contributed by AV, 16-Feb-2021.)

Assertion

Ref	Expression
elopabran	⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} → 𝐴 ∈ 𝑅)

Distinct variable groups:   𝑥,𝑅   𝑦,𝑅

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem elopabran

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝜓) → 𝑥𝑅𝑦)
2	1	ssopab2i 5569	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
3		opabss 5230	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
4	2, 3	sstri 4018	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ 𝑅
5	4	sseli 4004	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} → 𝐴 ∈ 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108   class class class wbr 5166  {copab 5228

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-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ss 3993  df-br 5167  df-opab 5229

This theorem is referenced by:  opabresex2  7502  fvmptopab  7504  clwlkwlk  29811