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Theorem disjiunel 29295
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 3572 . . . . . 6 (𝜑𝑌𝐴)
43snssd 4316 . . . . 5 (𝜑 → {𝑌} ⊆ 𝐴)
51, 4unssd 3773 . . . 4 (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴)
6 disjiunel.1 . . . 4 (𝜑Disj 𝑥𝐴 𝐵)
7 disjss1 4599 . . . 4 ((𝐸 ∪ {𝑌}) ⊆ 𝐴 → (Disj 𝑥𝐴 𝐵Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵))
85, 6, 7sylc 65 . . 3 (𝜑Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵)
92eldifbd 3573 . . . 4 (𝜑 → ¬ 𝑌𝐸)
10 disjiunel.2 . . . . 5 (𝑥 = 𝑌𝐵 = 𝐷)
1110disjunsn 29293 . . . 4 ((𝑌𝐴 ∧ ¬ 𝑌𝐸) → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥𝐸 𝐵 ∧ ( 𝑥𝐸 𝐵𝐷) = ∅)))
123, 9, 11syl2anc 692 . . 3 (𝜑 → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥𝐸 𝐵 ∧ ( 𝑥𝐸 𝐵𝐷) = ∅)))
138, 12mpbid 222 . 2 (𝜑 → (Disj 𝑥𝐸 𝐵 ∧ ( 𝑥𝐸 𝐵𝐷) = ∅))
1413simprd 479 1 (𝜑 → ( 𝑥𝐸 𝐵𝐷) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  cdif 3557  cun 3558  cin 3559  wss 3560  c0 3897  {csn 4155   ciun 4492  Disj wdisj 4593
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 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-sn 4156  df-iun 4494  df-disj 4594
This theorem is referenced by:  disjuniel  29296
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