Theorem dfopif 4797

Description: Rewrite df-op 4565 using if. When both arguments are sets, it reduces to the standard Kuratowski definition; otherwise, it is defined to be the empty set. Avoid directly depending on this detail so that theorems will not depend on the Kuratowski construction. (Contributed by Mario Carneiro, 26-Apr-2015.) Avoid ax-10 2139, ax-11 2156, ax-12 2173. (Revised by SN, 1-Aug-2024.) (Avoid depending on this detail.)

Assertion

Ref	Expression
dfopif	⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)

Proof of Theorem dfopif

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

Step	Hyp	Ref	Expression
1		noel 4261	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2	1	intnan 486	. . . . 5 ⊢ ¬ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ ∅)
3		biorf 933	. . . . 5 ⊢ (¬ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ ∅) → (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ ((¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ ∅) ∨ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))))
4	2, 3	ax-mp 5	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ ((¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ ∅) ∨ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})))
5		df-3an 1087	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
6		orcom 866	. . . 4 ⊢ ((((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ∨ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ ∅)) ↔ ((¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ ∅) ∨ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})))
7	4, 5, 6	3bitr4i 302	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ∨ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ ∅)))
8		eleq1 2826	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
9	8	3anbi3d 1440	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})))
10		df-op 4565	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
11	9, 10	elab2g 3604	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})))
12	11	elv 3428	. . 3 ⊢ (𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
13		elif 4499	. . 3 ⊢ (𝑥 ∈ if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) ↔ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ∨ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ ∅)))
14	7, 12, 13	3bitr4i 302	. 2 ⊢ (𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ 𝑥 ∈ if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅))
15	14	eqriv 2735	1 ⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ∅c0 4253  ifcif 4456  {csn 4558  {cpr 4560  ⟨cop 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-nul 4254  df-if 4457  df-op 4565

This theorem is referenced by:  dfopg  4799  opeq1  4801  opeq2  4802  nfop  4817  csbopg  4819  opprc  4824  opex  5373