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Theorem disjiunel 32609
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 3963 . . . . . 6 (𝜑𝑌𝐴)
43snssd 4809 . . . . 5 (𝜑 → {𝑌} ⊆ 𝐴)
51, 4unssd 4192 . . . 4 (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴)
6 disjiunel.1 . . . 4 (𝜑Disj 𝑥𝐴 𝐵)
7 disjss1 5116 . . . 4 ((𝐸 ∪ {𝑌}) ⊆ 𝐴 → (Disj 𝑥𝐴 𝐵Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵))
85, 6, 7sylc 65 . . 3 (𝜑Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵)
92eldifbd 3964 . . . 4 (𝜑 → ¬ 𝑌𝐸)
10 disjiunel.2 . . . . 5 (𝑥 = 𝑌𝐵 = 𝐷)
1110disjunsn 32607 . . . 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 1540  wcel 2108  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626   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-iun 4993  df-disj 5111
This theorem is referenced by:  disjuniel  32610
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