Theorem opabid2ss 30365

Description: One direction of opabid2 5697 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 22	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
2	1	opabssi 30364	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ⊆ 𝐴

Syntax hints:   ∈ wcel 2113   ⊆ wss 3933  ⟨cop 4570  {copab 5125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-in 3940  df-ss 3949  df-opab 5126

This theorem is referenced by: (None)