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Theorem disjiund 5069
Description: Conditions for a collection of index unions of sets 𝐴(𝑎, 𝑏) for 𝑎𝑉 and 𝑏𝑊 to be disjoint. (Contributed by AV, 9-Jan-2022.)
Hypotheses
Ref Expression
disjiund.1 (𝑎 = 𝑐𝐴 = 𝐶)
disjiund.2 (𝑏 = 𝑑𝐶 = 𝐷)
disjiund.3 (𝑎 = 𝑐𝑊 = 𝑋)
disjiund.4 ((𝜑𝑥𝐴𝑥𝐷) → 𝑎 = 𝑐)
Assertion
Ref Expression
disjiund (𝜑Disj 𝑎𝑉 𝑏𝑊 𝐴)
Distinct variable groups:   𝐴,𝑐,𝑑,𝑥   𝐶,𝑎,𝑑,𝑥   𝐷,𝑏   𝑉,𝑎,𝑐   𝑊,𝑏,𝑐,𝑑,𝑥   𝑋,𝑎,𝑏,𝑑,𝑥   𝜑,𝑎,𝑏,𝑐,𝑑,𝑥
Allowed substitution hints:   𝐴(𝑎,𝑏)   𝐶(𝑏,𝑐)   𝐷(𝑥,𝑎,𝑐,𝑑)   𝑉(𝑥,𝑏,𝑑)   𝑊(𝑎)   𝑋(𝑐)

Proof of Theorem disjiund
StepHypRef Expression
1 eliun 4934 . . . . . . . . 9 (𝑥 𝑏𝑊 𝐴 ↔ ∃𝑏𝑊 𝑥𝐴)
2 eliun 4934 . . . . . . . . . . . 12 (𝑥 𝑏𝑋 𝐶 ↔ ∃𝑏𝑋 𝑥𝐶)
3 disjiund.2 . . . . . . . . . . . . . . 15 (𝑏 = 𝑑𝐶 = 𝐷)
43eleq2d 2826 . . . . . . . . . . . . . 14 (𝑏 = 𝑑 → (𝑥𝐶𝑥𝐷))
54cbvrexvw 3382 . . . . . . . . . . . . 13 (∃𝑏𝑋 𝑥𝐶 ↔ ∃𝑑𝑋 𝑥𝐷)
6 disjiund.4 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑥𝐷) → 𝑎 = 𝑐)
763exp 1118 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑥𝐴 → (𝑥𝐷𝑎 = 𝑐)))
87rexlimdvw 3221 . . . . . . . . . . . . . . 15 (𝜑 → (∃𝑏𝑊 𝑥𝐴 → (𝑥𝐷𝑎 = 𝑐)))
98imp 407 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∃𝑏𝑊 𝑥𝐴) → (𝑥𝐷𝑎 = 𝑐))
109rexlimdvw 3221 . . . . . . . . . . . . 13 ((𝜑 ∧ ∃𝑏𝑊 𝑥𝐴) → (∃𝑑𝑋 𝑥𝐷𝑎 = 𝑐))
115, 10syl5bi 241 . . . . . . . . . . . 12 ((𝜑 ∧ ∃𝑏𝑊 𝑥𝐴) → (∃𝑏𝑋 𝑥𝐶𝑎 = 𝑐))
122, 11syl5bi 241 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑏𝑊 𝑥𝐴) → (𝑥 𝑏𝑋 𝐶𝑎 = 𝑐))
1312con3d 152 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑏𝑊 𝑥𝐴) → (¬ 𝑎 = 𝑐 → ¬ 𝑥 𝑏𝑋 𝐶))
1413impancom 452 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑎 = 𝑐) → (∃𝑏𝑊 𝑥𝐴 → ¬ 𝑥 𝑏𝑋 𝐶))
151, 14syl5bi 241 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑎 = 𝑐) → (𝑥 𝑏𝑊 𝐴 → ¬ 𝑥 𝑏𝑋 𝐶))
1615ralrimiv 3109 . . . . . . 7 ((𝜑 ∧ ¬ 𝑎 = 𝑐) → ∀𝑥 𝑏𝑊 𝐴 ¬ 𝑥 𝑏𝑋 𝐶)
17 disj 4387 . . . . . . 7 (( 𝑏𝑊 𝐴 𝑏𝑋 𝐶) = ∅ ↔ ∀𝑥 𝑏𝑊 𝐴 ¬ 𝑥 𝑏𝑋 𝐶)
1816, 17sylibr 233 . . . . . 6 ((𝜑 ∧ ¬ 𝑎 = 𝑐) → ( 𝑏𝑊 𝐴 𝑏𝑋 𝐶) = ∅)
1918ex 413 . . . . 5 (𝜑 → (¬ 𝑎 = 𝑐 → ( 𝑏𝑊 𝐴 𝑏𝑋 𝐶) = ∅))
2019orrd 860 . . . 4 (𝜑 → (𝑎 = 𝑐 ∨ ( 𝑏𝑊 𝐴 𝑏𝑋 𝐶) = ∅))
2120a1d 25 . . 3 (𝜑 → ((𝑎𝑉𝑐𝑉) → (𝑎 = 𝑐 ∨ ( 𝑏𝑊 𝐴 𝑏𝑋 𝐶) = ∅)))
2221ralrimivv 3116 . 2 (𝜑 → ∀𝑎𝑉𝑐𝑉 (𝑎 = 𝑐 ∨ ( 𝑏𝑊 𝐴 𝑏𝑋 𝐶) = ∅))
23 disjiund.3 . . 3 (𝑎 = 𝑐𝑊 = 𝑋)
24 disjiund.1 . . 3 (𝑎 = 𝑐𝐴 = 𝐶)
2523, 24disjiunb 5068 . 2 (Disj 𝑎𝑉 𝑏𝑊 𝐴 ↔ ∀𝑎𝑉𝑐𝑉 (𝑎 = 𝑐 ∨ ( 𝑏𝑊 𝐴 𝑏𝑋 𝐶) = ∅))
2622, 25sylibr 233 1 (𝜑Disj 𝑎𝑉 𝑏𝑊 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 844  w3a 1086   = wceq 1542  wcel 2110  wral 3066  wrex 3067  cin 3891  c0 4262   ciun 4930  Disj wdisj 5044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-mo 2542  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rmo 3074  df-v 3433  df-dif 3895  df-in 3899  df-ss 3909  df-nul 4263  df-iun 4932  df-disj 5045
This theorem is referenced by:  2wspiundisj  28324
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