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Theorem disjiunel 32616
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
disjiunel.1 (𝜑Disj 𝑥𝐴 𝐵)
disjiunel.2 (𝑥 = 𝑌𝐵 = 𝐷)
disjiunel.3 (𝜑𝐸𝐴)
disjiunel.4 (𝜑𝑌 ∈ (𝐴𝐸))
Assertion
Ref Expression
disjiunel (𝜑 → ( 𝑥𝐸 𝐵𝐷) = ∅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐷   𝑥,𝐸   𝑥,𝑌
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem disjiunel
StepHypRef Expression
1 disjiunel.3 . . . . 5 (𝜑𝐸𝐴)
2 disjiunel.4 . . . . . . 7 (𝜑𝑌 ∈ (𝐴𝐸))
32eldifad 3975 . . . . . 6 (𝜑𝑌𝐴)
43snssd 4814 . . . . 5 (𝜑 → {𝑌} ⊆ 𝐴)
51, 4unssd 4202 . . . 4 (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴)
6 disjiunel.1 . . . 4 (𝜑Disj 𝑥𝐴 𝐵)
7 disjss1 5121 . . . 4 ((𝐸 ∪ {𝑌}) ⊆ 𝐴 → (Disj 𝑥𝐴 𝐵Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵))
85, 6, 7sylc 65 . . 3 (𝜑Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵)
92eldifbd 3976 . . . 4 (𝜑 → ¬ 𝑌𝐸)
10 disjiunel.2 . . . . 5 (𝑥 = 𝑌𝐵 = 𝐷)
1110disjunsn 32614 . . . 4 ((𝑌𝐴 ∧ ¬ 𝑌𝐸) → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥𝐸 𝐵 ∧ ( 𝑥𝐸 𝐵𝐷) = ∅)))
123, 9, 11syl2anc 584 . . 3 (𝜑 → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥𝐸 𝐵 ∧ ( 𝑥𝐸 𝐵𝐷) = ∅)))
138, 12mpbid 232 . 2 (𝜑 → (Disj 𝑥𝐸 𝐵 ∧ ( 𝑥𝐸 𝐵𝐷) = ∅))
1413simprd 495 1 (𝜑 → ( 𝑥𝐸 𝐵𝐷) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  {csn 4631   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-iun 4998  df-disj 5116
This theorem is referenced by:  disjuniel  32617
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