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Theorem dfopif 4592
Description: Rewrite df-op 4377 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 4377 . 2 𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
2 df-3an 1102 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
32abbii 2923 . 2 {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
4 iftrue 4285 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = {{𝐴}, {𝐴, 𝐵}})
5 ibar 520 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})))
65abbi2dv 2926 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} = {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})})
74, 6eqtr2d 2841 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅))
8 pm2.21 121 . . . . . . 7 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑥 ∈ ∅))
98adantrd 481 . . . . . 6 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → 𝑥 ∈ ∅))
109abssdv 3873 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} ⊆ ∅)
11 ss0 4172 . . . . 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 4288 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = ∅)
1412, 13eqtr4d 2843 . . 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 176 . 2 {𝑥 ∣ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
161, 3, 153eqtri 2832 1 𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384  w3a 1100   = wceq 1637  wcel 2156  {cab 2792  Vcvv 3391  wss 3769  c0 4116  ifcif 4279  {csn 4370  {cpr 4372  cop 4376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-v 3393  df-dif 3772  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-op 4377
This theorem is referenced by:  dfopg  4593  opeq1  4595  opeq2  4596  nfop  4611  csbopg  4613  opprc  4618  opex  5122
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