Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjorf Structured version   Visualization version   GIF version

Theorem disjorf 29730
Description: Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
Hypotheses
Ref Expression
disjorf.1 𝑖𝐴
disjorf.2 𝑗𝐴
disjorf.3 (𝑖 = 𝑗𝐵 = 𝐶)
Assertion
Ref Expression
disjorf (Disj 𝑖𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅))
Distinct variable groups:   𝑖,𝑗   𝐵,𝑗   𝐶,𝑖
Allowed substitution hints:   𝐴(𝑖,𝑗)   𝐵(𝑖)   𝐶(𝑗)

Proof of Theorem disjorf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-disj 4755 . 2 (Disj 𝑖𝐴 𝐵 ↔ ∀𝑥∃*𝑖𝐴 𝑥𝐵)
2 ralcom4 3376 . . 3 (∀𝑖𝐴𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
3 orcom 857 . . . . . . 7 ((𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ((𝐵𝐶) = ∅ ∨ 𝑖 = 𝑗))
4 df-or 835 . . . . . . 7 (((𝐵𝐶) = ∅ ∨ 𝑖 = 𝑗) ↔ (¬ (𝐵𝐶) = ∅ → 𝑖 = 𝑗))
5 neq0 4077 . . . . . . . . . 10 (¬ (𝐵𝐶) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵𝐶))
6 elin 3947 . . . . . . . . . . 11 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
76exbii 1924 . . . . . . . . . 10 (∃𝑥 𝑥 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐵𝑥𝐶))
85, 7bitri 264 . . . . . . . . 9 (¬ (𝐵𝐶) = ∅ ↔ ∃𝑥(𝑥𝐵𝑥𝐶))
98imbi1i 338 . . . . . . . 8 ((¬ (𝐵𝐶) = ∅ → 𝑖 = 𝑗) ↔ (∃𝑥(𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
10 19.23v 2023 . . . . . . . 8 (∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) ↔ (∃𝑥(𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
119, 10bitr4i 267 . . . . . . 7 ((¬ (𝐵𝐶) = ∅ → 𝑖 = 𝑗) ↔ ∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
123, 4, 113bitri 286 . . . . . 6 ((𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
1312ralbii 3129 . . . . 5 (∀𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ∀𝑗𝐴𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
14 ralcom4 3376 . . . . 5 (∀𝑗𝐴𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
1513, 14bitri 264 . . . 4 (∀𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ∀𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
1615ralbii 3129 . . 3 (∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ∀𝑖𝐴𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
17 disjorf.1 . . . . 5 𝑖𝐴
18 disjorf.2 . . . . 5 𝑗𝐴
19 nfv 1995 . . . . 5 𝑖 𝑥𝐶
20 disjorf.3 . . . . . 6 (𝑖 = 𝑗𝐵 = 𝐶)
2120eleq2d 2836 . . . . 5 (𝑖 = 𝑗 → (𝑥𝐵𝑥𝐶))
2217, 18, 19, 21rmo4f 29676 . . . 4 (∃*𝑖𝐴 𝑥𝐵 ↔ ∀𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
2322albii 1895 . . 3 (∀𝑥∃*𝑖𝐴 𝑥𝐵 ↔ ∀𝑥𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
242, 16, 233bitr4i 292 . 2 (∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ∀𝑥∃*𝑖𝐴 𝑥𝐵)
251, 24bitr4i 267 1 (Disj 𝑖𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 834  wal 1629   = wceq 1631  wex 1852  wcel 2145  wnfc 2900  wral 3061  ∃*wrmo 3064  cin 3722  c0 4063  Disj wdisj 4754
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 835  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-rmo 3069  df-v 3353  df-dif 3726  df-in 3730  df-nul 4064  df-disj 4755
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator