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Theorem disjiunel 30330
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 3924 . . . . . 6 (𝜑𝑌𝐴)
43snssd 4716 . . . . 5 (𝜑 → {𝑌} ⊆ 𝐴)
51, 4unssd 4138 . . . 4 (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴)
6 disjiunel.1 . . . 4 (𝜑Disj 𝑥𝐴 𝐵)
7 disjss1 5011 . . . 4 ((𝐸 ∪ {𝑌}) ⊆ 𝐴 → (Disj 𝑥𝐴 𝐵Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵))
85, 6, 7sylc 65 . . 3 (𝜑Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵)
92eldifbd 3925 . . . 4 (𝜑 → ¬ 𝑌𝐸)
10 disjiunel.2 . . . . 5 (𝑥 = 𝑌𝐵 = 𝐷)
1110disjunsn 30328 . . . 4 ((𝑌𝐴 ∧ ¬ 𝑌𝐸) → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥𝐸 𝐵 ∧ ( 𝑥𝐸 𝐵𝐷) = ∅)))
123, 9, 11syl2anc 586 . . 3 (𝜑 → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥𝐸 𝐵 ∧ ( 𝑥𝐸 𝐵𝐷) = ∅)))
138, 12mpbid 234 . 2 (𝜑 → (Disj 𝑥𝐸 𝐵 ∧ ( 𝑥𝐸 𝐵𝐷) = ∅))
1413simprd 498 1 (𝜑 → ( 𝑥𝐸 𝐵𝐷) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  cdif 3909  cun 3910  cin 3911  wss 3912  c0 4267  {csn 4541   ciun 4893  Disj wdisj 5005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-sn 4542  df-iun 4895  df-disj 5006
This theorem is referenced by:  disjuniel  30331
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