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Theorem dfopif 4875
Description: Rewrite df-op 4638 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.
StepHypRef Expression
1 df-op 4638 . 2 𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
2 df-3an 1088 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
32abbii 2807 . 2 {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
4 iftrue 4537 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = {{𝐴}, {𝐴, 𝐵}})
5 ibar 528 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})))
65eqabdv 2873 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} = {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})})
74, 6eqtr2d 2776 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅))
8 pm2.21 123 . . . . . . 7 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑥 ∈ ∅))
98adantrd 491 . . . . . 6 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → 𝑥 ∈ ∅))
109abssdv 4078 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} ⊆ ∅)
11 ss0 4408 . . . . 5 ({𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} ⊆ ∅ → {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = ∅)
1210, 11syl 17 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = ∅)
13 iffalse 4540 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = ∅)
1412, 13eqtr4d 2778 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅))
157, 14pm2.61i 182 . 2 {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
161, 3, 153eqtri 2767 1 𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  Vcvv 3478  wss 3963  c0 4339  ifcif 4531  {csn 4631  {cpr 4633  cop 4637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-dif 3966  df-ss 3980  df-nul 4340  df-if 4532  df-op 4638
This theorem is referenced by:  dfopg  4876  opeq1  4878  opeq2  4879  nfop  4894  csbopg  4896  opprc  4901  opex  5475
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