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Theorem disjuniel 29255
 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 4539 . . 3 𝐵 = 𝑥𝐵 𝑥
21ineq1i 3788 . 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 29254 . 2 (𝜑 → ( 𝑥𝐵 𝑥𝐶) = ∅)
82, 7syl5eq 2667 1 (𝜑 → ( 𝐵𝐶) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987   ∖ cdif 3552   ∩ cin 3554   ⊆ wss 3555  ∅c0 3891  ∪ cuni 4402  ∪ ciun 4485  Disj wdisj 4583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-uni 4403  df-iun 4487  df-disj 4584 This theorem is referenced by:  carsgclctunlem1  30160
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