Theorem bj-opelidb 37993

Description: Characterization of the ordered pair elements of the identity relation.

Remark: in deduction-style proofs, one could save a few syntactic steps by using another antecedent than ⊤ which already appears in the proof. Here for instance this could be the definition I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} but this would make the proof less easy to read. (Contributed by BJ, 27-Dec-2023.)

Assertion

Ref	Expression
bj-opelidb	⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))

Proof of Theorem bj-opelidb

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-id 5542	. . . 4 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
2	1	a1i 11	. . 3 ⊢ (⊤ → I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
3		eqeq12 2777	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
4	3	adantl 487	. . 3 ⊢ ((⊤ ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
5	2, 4	opelopabbv 37984	. 2 ⊢ (⊤ → (⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)))
6	5	mptru 1577	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145  Vcvv 3450  ⟨cop 4589  {copab 5166   I cid 5541

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-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-opab 5167  df-id 5542

This theorem is used by:  bj-opelidb1  37994  bj-opelid  37997