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Theorem disjuniel 32617
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 5063 . . 3 𝐵 = 𝑥𝐵 𝑥
21ineq1i 4224 . 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 32616 . 2 (𝜑 → ( 𝑥𝐵 𝑥𝐶) = ∅)
82, 7eqtrid 2787 1 (𝜑 → ( 𝐵𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cdif 3960  cin 3962  wss 3963  c0 4339   cuni 4912   ciun 4996  Disj wdisj 5115
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-10 2139  ax-11 2155  ax-12 2175  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-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-uni 4913  df-iun 4998  df-disj 5116
This theorem is referenced by:  carsgclctunlem1  34299
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