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Theorem disjnim 4009
Description: If a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Jim Kingdon, 6-Oct-2022.)
Hypothesis
Ref Expression
disjnim.1 (𝑖 = 𝑗𝐵 = 𝐶)
Assertion
Ref Expression
disjnim (Disj 𝑖𝐴 𝐵 → ∀𝑖𝐴𝑗𝐴 (𝑖𝑗 → (𝐵𝐶) = ∅))
Distinct variable groups:   𝑖,𝑗,𝐴   𝐵,𝑗   𝐶,𝑖
Allowed substitution hints:   𝐵(𝑖)   𝐶(𝑗)

Proof of Theorem disjnim
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-disj 3996 . 2 (Disj 𝑖𝐴 𝐵 ↔ ∀𝑥∃*𝑖𝐴 𝑥𝐵)
2 disjnim.1 . . . . . . 7 (𝑖 = 𝑗𝐵 = 𝐶)
32eleq2d 2259 . . . . . 6 (𝑖 = 𝑗 → (𝑥𝐵𝑥𝐶))
43rmo4 2945 . . . . 5 (∃*𝑖𝐴 𝑥𝐵 ↔ ∀𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
54albii 1481 . . . 4 (∀𝑥∃*𝑖𝐴 𝑥𝐵 ↔ ∀𝑥𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
6 ralcom4 2774 . . . 4 (∀𝑖𝐴𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
75, 6bitr4i 187 . . 3 (∀𝑥∃*𝑖𝐴 𝑥𝐵 ↔ ∀𝑖𝐴𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
8 ralcom4 2774 . . . . 5 (∀𝑗𝐴𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
9 19.23v 1894 . . . . . . . . 9 (∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) ↔ (∃𝑥(𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
109biimpi 120 . . . . . . . 8 (∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) → (∃𝑥(𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
1110necon3ad 2402 . . . . . . 7 (∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) → (𝑖𝑗 → ¬ ∃𝑥(𝑥𝐵𝑥𝐶)))
12 notm0 3458 . . . . . . . 8 (¬ ∃𝑥 𝑥 ∈ (𝐵𝐶) ↔ (𝐵𝐶) = ∅)
13 elin 3333 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1413exbii 1616 . . . . . . . . 9 (∃𝑥 𝑥 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐵𝑥𝐶))
1514notbii 669 . . . . . . . 8 (¬ ∃𝑥 𝑥 ∈ (𝐵𝐶) ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐶))
1612, 15bitr3i 186 . . . . . . 7 ((𝐵𝐶) = ∅ ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐶))
1711, 16imbitrrdi 162 . . . . . 6 (∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) → (𝑖𝑗 → (𝐵𝐶) = ∅))
1817ralimi 2553 . . . . 5 (∀𝑗𝐴𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) → ∀𝑗𝐴 (𝑖𝑗 → (𝐵𝐶) = ∅))
198, 18sylbir 135 . . . 4 (∀𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) → ∀𝑗𝐴 (𝑖𝑗 → (𝐵𝐶) = ∅))
2019ralimi 2553 . . 3 (∀𝑖𝐴𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) → ∀𝑖𝐴𝑗𝐴 (𝑖𝑗 → (𝐵𝐶) = ∅))
217, 20sylbi 121 . 2 (∀𝑥∃*𝑖𝐴 𝑥𝐵 → ∀𝑖𝐴𝑗𝐴 (𝑖𝑗 → (𝐵𝐶) = ∅))
221, 21sylbi 121 1 (Disj 𝑖𝐴 𝐵 → ∀𝑖𝐴𝑗𝐴 (𝑖𝑗 → (𝐵𝐶) = ∅))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wal 1362   = wceq 1364  wex 1503  wcel 2160  wne 2360  wral 2468  ∃*wrmo 2471  cin 3143  c0 3437  Disj wdisj 3995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rmo 2476  df-v 2754  df-dif 3146  df-in 3150  df-nul 3438  df-disj 3996
This theorem is referenced by:  disjnims  4010
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