Theorem elopabran 5540

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 488	. . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝜓) → 𝑥𝑅𝑦)
2	1	ssopab2i 5529	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
3		opabss 5172	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
4	2, 3	sstri 3922	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ 𝑅
5	4	sseli 3909	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} → 𝐴 ∈ 𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147   class class class wbr 5106  {copab 5170

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ss 3901  df-br 5107  df-opab 5171

This theorem is used by:  opabresex2  7494  fvmptopab  7495  clwlkwlk  30389