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Theorem disjuniel 29748
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 4708 . . 3 𝐵 = 𝑥𝐵 𝑥
21ineq1i 3961 . 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 29747 . 2 (𝜑 → ( 𝑥𝐵 𝑥𝐶) = ∅)
82, 7syl5eq 2817 1 (𝜑 → ( 𝐵𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cdif 3720  cin 3722  wss 3723  c0 4063   cuni 4575   ciun 4655  Disj wdisj 4755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-sn 4318  df-uni 4576  df-iun 4657  df-disj 4756
This theorem is referenced by:  carsgclctunlem1  30719
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