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Theorem disjiunel 28791
 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 3552 . . . . . 6 (𝜑𝑌𝐴)
43snssd 4281 . . . . 5 (𝜑 → {𝑌} ⊆ 𝐴)
51, 4unssd 3751 . . . 4 (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴)
6 disjiunel.1 . . . 4 (𝜑Disj 𝑥𝐴 𝐵)
7 disjss1 4559 . . . . 5 ((𝐸 ∪ {𝑌}) ⊆ 𝐴 → (Disj 𝑥𝐴 𝐵Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵))
87imp 444 . . . 4 (((𝐸 ∪ {𝑌}) ⊆ 𝐴Disj 𝑥𝐴 𝐵) → Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵)
95, 6, 8syl2anc 691 . . 3 (𝜑Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵)
102eldifbd 3553 . . . 4 (𝜑 → ¬ 𝑌𝐸)
11 disjiunel.2 . . . . 5 (𝑥 = 𝑌𝐵 = 𝐷)
1211disjunsn 28789 . . . 4 ((𝑌𝐴 ∧ ¬ 𝑌𝐸) → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥𝐸 𝐵 ∧ ( 𝑥𝐸 𝐵𝐷) = ∅)))
133, 10, 12syl2anc 691 . . 3 (𝜑 → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥𝐸 𝐵 ∧ ( 𝑥𝐸 𝐵𝐷) = ∅)))
149, 13mpbid 221 . 2 (𝜑 → (Disj 𝑥𝐸 𝐵 ∧ ( 𝑥𝐸 𝐵𝐷) = ∅))
1514simprd 478 1 (𝜑 → ( 𝑥𝐸 𝐵𝐷) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ ciun 4455  Disj wdisj 4553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-iun 4457  df-disj 4554 This theorem is referenced by:  disjuniel  28792
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