Theorem dfopif 4755

Description: Rewrite df-op 4523 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 2145, ax-11 2162, ax-12 2179. (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 4219	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2	1	intnan 490	. . . . 5 ⊢ ¬ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ ∅)
3		biorf 936	. . . . 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 1090	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
6		orcom 869	. . . 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 306	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ∨ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ ∅)))
8		eleq1 2820	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
9	8	3anbi3d 1443	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})))
10		df-op 4523	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
11	9, 10	elab2g 3575	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})))
12	11	elv 3404	. . 3 ⊢ (𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
13		elif 4457	. . 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 306	. 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 209   ∧ wa 399   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  Vcvv 3398  ∅c0 4211  ifcif 4414  {csn 4516  {cpr 4518  ⟨cop 4522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-dif 3846  df-nul 4212  df-if 4415  df-op 4523

This theorem is referenced by:  dfopg  4757  opeq1OLD  4760  opeq2OLD  4762  nfop  4777  csbopg  4779  opprc  4784  opex  5322