Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  disjnim GIF version

Theorem disjnim 3920
 Description: If a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 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 3907 . 2 (Disj 𝑖𝐴 𝐵 ↔ ∀𝑥∃*𝑖𝐴 𝑥𝐵)
2 disjnim.1 . . . . . . 7 (𝑖 = 𝑗𝐵 = 𝐶)
32eleq2d 2209 . . . . . 6 (𝑖 = 𝑗 → (𝑥𝐵𝑥𝐶))
43rmo4 2877 . . . . 5 (∃*𝑖𝐴 𝑥𝐵 ↔ ∀𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
54albii 1446 . . . 4 (∀𝑥∃*𝑖𝐴 𝑥𝐵 ↔ ∀𝑥𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
6 ralcom4 2708 . . . 4 (∀𝑖𝐴𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
75, 6bitr4i 186 . . 3 (∀𝑥∃*𝑖𝐴 𝑥𝐵 ↔ ∀𝑖𝐴𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
8 ralcom4 2708 . . . . 5 (∀𝑗𝐴𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
9 19.23v 1855 . . . . . . . . 9 (∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) ↔ (∃𝑥(𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
109biimpi 119 . . . . . . . 8 (∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) → (∃𝑥(𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
1110necon3ad 2350 . . . . . . 7 (∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) → (𝑖𝑗 → ¬ ∃𝑥(𝑥𝐵𝑥𝐶)))
12 notm0 3383 . . . . . . . 8 (¬ ∃𝑥 𝑥 ∈ (𝐵𝐶) ↔ (𝐵𝐶) = ∅)
13 elin 3259 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1413exbii 1584 . . . . . . . . 9 (∃𝑥 𝑥 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐵𝑥𝐶))
1514notbii 657 . . . . . . . 8 (¬ ∃𝑥 𝑥 ∈ (𝐵𝐶) ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐶))
1612, 15bitr3i 185 . . . . . . 7 ((𝐵𝐶) = ∅ ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐶))
1711, 16syl6ibr 161 . . . . . 6 (∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) → (𝑖𝑗 → (𝐵𝐶) = ∅))
1817ralimi 2495 . . . . 5 (∀𝑗𝐴𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) → ∀𝑗𝐴 (𝑖𝑗 → (𝐵𝐶) = ∅))
198, 18sylbir 134 . . . 4 (∀𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) → ∀𝑗𝐴 (𝑖𝑗 → (𝐵𝐶) = ∅))
2019ralimi 2495 . . 3 (∀𝑖𝐴𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) → ∀𝑖𝐴𝑗𝐴 (𝑖𝑗 → (𝐵𝐶) = ∅))
217, 20sylbi 120 . 2 (∀𝑥∃*𝑖𝐴 𝑥𝐵 → ∀𝑖𝐴𝑗𝐴 (𝑖𝑗 → (𝐵𝐶) = ∅))
221, 21sylbi 120 1 (Disj 𝑖𝐴 𝐵 → ∀𝑖𝐴𝑗𝐴 (𝑖𝑗 → (𝐵𝐶) = ∅))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480   ≠ wne 2308  ∀wral 2416  ∃*wrmo 2419   ∩ cin 3070  ∅c0 3363  Disj wdisj 3906 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rmo 2424  df-v 2688  df-dif 3073  df-in 3077  df-nul 3364  df-disj 3907 This theorem is referenced by:  disjnims  3921
 Copyright terms: Public domain W3C validator