Theorem dfopif 4836

Description: Rewrite df-op 4598 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 depending on this detail.)

Assertion

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

Proof of Theorem dfopif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-op 4598	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
2		df-3an 1103	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
3	2	abbii 2836	. 2 ⊢ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
4		iftrue 4495	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = {{𝐴}, {𝐴, 𝐵}})
5		ibar 537	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})))
6	5	eqabdv 2902	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} = {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})})
7	4, 6	eqtr2d 2805	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅))
8		pm2.21 124	. . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑥 ∈ ∅))
9	8	adantrd 496	. . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → 𝑥 ∈ ∅))
10	9	abssdv 4029	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} ⊆ ∅)
11		ss0 4365	. . . . 5 ⊢ ({𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} ⊆ ∅ → {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = ∅)
12	10, 11	syl 18	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = ∅)
13		iffalse 4498	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = ∅)
14	12, 13	eqtr4d 2807	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅))
15	7, 14	pm2.61i 184	. 2 ⊢ {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
16	1, 3, 15	3eqtri 2796	1 ⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)

Syntax hints:  ¬ wn 3   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  {cab 2747  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  ifcif 4489  {csn 4591  {cpr 4593  ⟨cop 4597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-dif 3916  df-ss 3930  df-nul 4295  df-if 4490  df-op 4598

This theorem is referenced by:  dfopg  4837  opeq1  4839  opeq2  4840  nfop  4855  csbopg  4857  opprc  4862  opexOLD  5444