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Theorem disjuniel 30107
Description: A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.)
Hypotheses
Ref Expression
disjuniel.1 (𝜑Disj 𝑥𝐴 𝑥)
disjuniel.2 (𝜑𝐵𝐴)
disjuniel.3 (𝜑𝐶 ∈ (𝐴𝐵))
Assertion
Ref Expression
disjuniel (𝜑 → ( 𝐵𝐶) = ∅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem disjuniel
StepHypRef Expression
1 uniiun 4842 . . 3 𝐵 = 𝑥𝐵 𝑥
21ineq1i 4066 . 2 ( 𝐵𝐶) = ( 𝑥𝐵 𝑥𝐶)
3 disjuniel.1 . . 3 (𝜑Disj 𝑥𝐴 𝑥)
4 id 22 . . 3 (𝑥 = 𝐶𝑥 = 𝐶)
5 disjuniel.2 . . 3 (𝜑𝐵𝐴)
6 disjuniel.3 . . 3 (𝜑𝐶 ∈ (𝐴𝐵))
73, 4, 5, 6disjiunel 30106 . 2 (𝜑 → ( 𝑥𝐵 𝑥𝐶) = ∅)
82, 7syl5eq 2820 1 (𝜑 → ( 𝐵𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  cdif 3820  cin 3822  wss 3823  c0 4172   cuni 4706   ciun 4786  Disj wdisj 4891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-sn 4436  df-uni 4707  df-iun 4788  df-disj 4892
This theorem is referenced by:  carsgclctunlem1  31220
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