Theorem ideqg 5837

Description: For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
ideqg	⊢ (𝐵 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem ideqg

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

Step	Hyp	Ref	Expression
1		id 23	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ 𝑉)
2		reli 5813	. . . 4 ⊢ Rel I
3	2	brrelex1i 5717	. . 3 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V)
4	1, 3	anim12ci 625	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝑉))
5		eleq1 2851	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
6	5	biimparc 484	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑉)
7	6	elexd 3478	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
8		simpl 487	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑉)
9	7, 8	jca 520	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝑉))
10		eqeq1 2767	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
11		eqeq2 2775	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵))
12		df-id 5556	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
13	10, 11, 12	brabg 5524	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
14	4, 9, 13	pm5.21nd 813	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   class class class wbr 5109   I cid 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668

This theorem is referenced by:  ideq  5838  ididg  5839  poleloe  6131  isof1oidb  7322  pltval  18381  tglngne  28819  tgelrnln  28903  opeldifid  32944  ideq2  38982  idinxpss  38987  inxpssidinxp  38991  idinxpssinxp  38992  cnvref5  39020  rnxrnidres  39093  dfsucmap3  39132  cossid  39239  fourierdlem42  46883