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Theorem inpr0 30631
Description: Rewrite an empty intersection with a pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
Assertion
Ref Expression
inpr0 ((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ (¬ 𝐵𝐴 ∧ ¬ 𝐶𝐴))

Proof of Theorem inpr0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 r19.26 3095 . 2 (∀𝑥𝐴 (𝑥𝐵𝑥𝐶) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 𝑥𝐶))
2 nelpr 30630 . . . . . 6 (𝑥 ∈ V → (¬ 𝑥 ∈ {𝐵, 𝐶} ↔ (𝑥𝐵𝑥𝐶)))
32elv 3429 . . . . 5 𝑥 ∈ {𝐵, 𝐶} ↔ (𝑥𝐵𝑥𝐶))
43imbi2i 339 . . . 4 ((𝑥𝐴 → ¬ 𝑥 ∈ {𝐵, 𝐶}) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
54albii 1827 . . 3 (∀𝑥(𝑥𝐴 → ¬ 𝑥 ∈ {𝐵, 𝐶}) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
6 disj1 4382 . . 3 ((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥 ∈ {𝐵, 𝐶}))
7 df-ral 3069 . . 3 (∀𝑥𝐴 (𝑥𝐵𝑥𝐶) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
85, 6, 73bitr4i 306 . 2 ((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ ∀𝑥𝐴 (𝑥𝐵𝑥𝐶))
9 nelb 3195 . . 3 𝐵𝐴 ↔ ∀𝑥𝐴 𝑥𝐵)
10 nelb 3195 . . 3 𝐶𝐴 ↔ ∀𝑥𝐴 𝑥𝐶)
119, 10anbi12i 630 . 2 ((¬ 𝐵𝐴 ∧ ¬ 𝐶𝐴) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 𝑥𝐶))
121, 8, 113bitr4i 306 1 ((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ (¬ 𝐵𝐴 ∧ ¬ 𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1541   = wceq 1543  wcel 2112  wne 2943  wral 3064  Vcvv 3423  cin 3882  c0 4254  {cpr 4560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ne 2944  df-ral 3069  df-rex 3070  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-nul 4255  df-sn 4559  df-pr 4561
This theorem is referenced by: (None)
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