Theorem elopabran 5532

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 5521	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
3		opabss 5168	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
4	2, 3	sstri 3939	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ 𝑅
5	4	sseli 3926	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} → 𝐴 ∈ 𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145   class class class wbr 5102  {copab 5166

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ss 3915  df-br 5103  df-opab 5167

This theorem is used by:  opabresex2  7462  fvmptopab  7463  clwlkwlk  30295