Theorem opabid2ss 33088

Description: One direction of opabid2 5809 which holds without a Rel 𝐴 requirement. (Contributed by Thierry Arnoux, 18-Feb-2022.)

Assertion

Ref	Expression
opabid2ss	⊢ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ⊆ 𝐴

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem opabid2ss

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
2	1	opabssi 33087	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ⊆ 𝐴

This proof depends on syntax axioms:   ∈ wcel 2145   ⊆ wss 3899  ⟨cop 4590  {copab 5167

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 3916  df-opab 5168

This theorem is used by: (None)