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Theorem disjiund 4778
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 4659 . . . . . . . . 9 (𝑥 𝑏𝑊 𝐴 ↔ ∃𝑏𝑊 𝑥𝐴)
2 eliun 4659 . . . . . . . . . . . 12 (𝑥 𝑏𝑋 𝐶 ↔ ∃𝑏𝑋 𝑥𝐶)
3 disjiund.2 . . . . . . . . . . . . . . 15 (𝑏 = 𝑑𝐶 = 𝐷)
43eleq2d 2836 . . . . . . . . . . . . . 14 (𝑏 = 𝑑 → (𝑥𝐶𝑥𝐷))
54cbvrexv 3321 . . . . . . . . . . . . 13 (∃𝑏𝑋 𝑥𝐶 ↔ ∃𝑑𝑋 𝑥𝐷)
6 disjiund.4 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑥𝐷) → 𝑎 = 𝑐)
763exp 1112 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑥𝐴 → (𝑥𝐷𝑎 = 𝑐)))
87rexlimdvw 3182 . . . . . . . . . . . . . . 15 (𝜑 → (∃𝑏𝑊 𝑥𝐴 → (𝑥𝐷𝑎 = 𝑐)))
98imp 393 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∃𝑏𝑊 𝑥𝐴) → (𝑥𝐷𝑎 = 𝑐))
109rexlimdvw 3182 . . . . . . . . . . . . 13 ((𝜑 ∧ ∃𝑏𝑊 𝑥𝐴) → (∃𝑑𝑋 𝑥𝐷𝑎 = 𝑐))
115, 10syl5bi 232 . . . . . . . . . . . 12 ((𝜑 ∧ ∃𝑏𝑊 𝑥𝐴) → (∃𝑏𝑋 𝑥𝐶𝑎 = 𝑐))
122, 11syl5bi 232 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑏𝑊 𝑥𝐴) → (𝑥 𝑏𝑋 𝐶𝑎 = 𝑐))
1312con3d 149 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑏𝑊 𝑥𝐴) → (¬ 𝑎 = 𝑐 → ¬ 𝑥 𝑏𝑋 𝐶))
1413impancom 439 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑎 = 𝑐) → (∃𝑏𝑊 𝑥𝐴 → ¬ 𝑥 𝑏𝑋 𝐶))
151, 14syl5bi 232 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑎 = 𝑐) → (𝑥 𝑏𝑊 𝐴 → ¬ 𝑥 𝑏𝑋 𝐶))
1615ralrimiv 3114 . . . . . . 7 ((𝜑 ∧ ¬ 𝑎 = 𝑐) → ∀𝑥 𝑏𝑊 𝐴 ¬ 𝑥 𝑏𝑋 𝐶)
17 disj 4161 . . . . . . 7 (( 𝑏𝑊 𝐴 𝑏𝑋 𝐶) = ∅ ↔ ∀𝑥 𝑏𝑊 𝐴 ¬ 𝑥 𝑏𝑋 𝐶)
1816, 17sylibr 224 . . . . . 6 ((𝜑 ∧ ¬ 𝑎 = 𝑐) → ( 𝑏𝑊 𝐴 𝑏𝑋 𝐶) = ∅)
1918ex 397 . . . . 5 (𝜑 → (¬ 𝑎 = 𝑐 → ( 𝑏𝑊 𝐴 𝑏𝑋 𝐶) = ∅))
2019orrd 844 . . . 4 (𝜑 → (𝑎 = 𝑐 ∨ ( 𝑏𝑊 𝐴 𝑏𝑋 𝐶) = ∅))
2120a1d 25 . . 3 (𝜑 → ((𝑎𝑉𝑐𝑉) → (𝑎 = 𝑐 ∨ ( 𝑏𝑊 𝐴 𝑏𝑋 𝐶) = ∅)))
2221ralrimivv 3119 . 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 4777 . 2 (Disj 𝑎𝑉 𝑏𝑊 𝐴 ↔ ∀𝑎𝑉𝑐𝑉 (𝑎 = 𝑐 ∨ ( 𝑏𝑊 𝐴 𝑏𝑋 𝐶) = ∅))
2622, 25sylibr 224 1 (𝜑Disj 𝑎𝑉 𝑏𝑊 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wo 828  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  cin 3723  c0 4064   ciun 4655  Disj wdisj 4755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rmo 3069  df-v 3353  df-dif 3727  df-in 3731  df-ss 3738  df-nul 4065  df-iun 4657  df-disj 4756
This theorem is referenced by:  2wspiundisj  27113
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