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Theorem disjord 5051
Description: Conditions for a collection of sets 𝐴(𝑎) for 𝑎𝑉 to be disjoint. (Contributed by AV, 9-Jan-2022.)
Hypotheses
Ref Expression
disjord.1 (𝑎 = 𝑏𝐴 = 𝐵)
disjord.2 ((𝜑𝑥𝐴𝑥𝐵) → 𝑎 = 𝑏)
Assertion
Ref Expression
disjord (𝜑Disj 𝑎𝑉 𝐴)
Distinct variable groups:   𝐴,𝑏,𝑥   𝐵,𝑎,𝑥   𝑉,𝑎,𝑏,𝑥   𝜑,𝑎,𝑏,𝑥
Allowed substitution hints:   𝐴(𝑎)   𝐵(𝑏)

Proof of Theorem disjord
StepHypRef Expression
1 orc 863 . . . . . 6 (𝑎 = 𝑏 → (𝑎 = 𝑏 ∨ (𝐴𝐵) = ∅))
21a1d 25 . . . . 5 (𝑎 = 𝑏 → (𝜑 → (𝑎 = 𝑏 ∨ (𝐴𝐵) = ∅)))
3 disjord.2 . . . . . . . . . . . 12 ((𝜑𝑥𝐴𝑥𝐵) → 𝑎 = 𝑏)
433expia 1115 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑥𝐵𝑎 = 𝑏))
54con3d 155 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (¬ 𝑎 = 𝑏 → ¬ 𝑥𝐵))
65impancom 452 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑎 = 𝑏) → (𝑥𝐴 → ¬ 𝑥𝐵))
76ralrimiv 3186 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑎 = 𝑏) → ∀𝑥𝐴 ¬ 𝑥𝐵)
8 disj 4402 . . . . . . . 8 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
97, 8sylibr 235 . . . . . . 7 ((𝜑 ∧ ¬ 𝑎 = 𝑏) → (𝐴𝐵) = ∅)
109olcd 872 . . . . . 6 ((𝜑 ∧ ¬ 𝑎 = 𝑏) → (𝑎 = 𝑏 ∨ (𝐴𝐵) = ∅))
1110expcom 414 . . . . 5 𝑎 = 𝑏 → (𝜑 → (𝑎 = 𝑏 ∨ (𝐴𝐵) = ∅)))
122, 11pm2.61i 183 . . . 4 (𝜑 → (𝑎 = 𝑏 ∨ (𝐴𝐵) = ∅))
1312adantr 481 . . 3 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎 = 𝑏 ∨ (𝐴𝐵) = ∅))
1413ralrimivva 3196 . 2 (𝜑 → ∀𝑎𝑉𝑏𝑉 (𝑎 = 𝑏 ∨ (𝐴𝐵) = ∅))
15 disjord.1 . . 3 (𝑎 = 𝑏𝐴 = 𝐵)
1615disjor 5043 . 2 (Disj 𝑎𝑉 𝐴 ↔ ∀𝑎𝑉𝑏𝑉 (𝑎 = 𝑏 ∨ (𝐴𝐵) = ∅))
1714, 16sylibr 235 1 (𝜑Disj 𝑎𝑉 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 843  w3a 1081   = wceq 1530  wcel 2107  wral 3143  cin 3939  c0 4295  Disj wdisj 5028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rmo 3151  df-v 3502  df-dif 3943  df-in 3947  df-nul 4296  df-disj 5029
This theorem is referenced by:  2wspdisj  27658
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