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Theorem disjuniel 32610
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 5058 . . 3 𝐵 = 𝑥𝐵 𝑥
21ineq1i 4216 . 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 32609 . 2 (𝜑 → ( 𝑥𝐵 𝑥𝐶) = ∅)
82, 7eqtrid 2789 1 (𝜑 → ( 𝐵𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cdif 3948  cin 3950  wss 3951  c0 4333   cuni 4907   ciun 4991  Disj wdisj 5110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-uni 4908  df-iun 4993  df-disj 5111
This theorem is referenced by:  carsgclctunlem1  34319
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