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Theorem disjor 5037
Description: Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
Hypothesis
Ref Expression
disjor.1 (𝑖 = 𝑗𝐵 = 𝐶)
Assertion
Ref Expression
disjor (Disj 𝑖𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅))
Distinct variable groups:   𝑖,𝑗,𝐴   𝐵,𝑗   𝐶,𝑖
Allowed substitution hints:   𝐵(𝑖)   𝐶(𝑗)

Proof of Theorem disjor
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-disj 5023 . 2 (Disj 𝑖𝐴 𝐵 ↔ ∀𝑥∃*𝑖𝐴 𝑥𝐵)
2 ralcom4 3232 . . 3 (∀𝑖𝐴𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
3 orcom 864 . . . . . . 7 ((𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ((𝐵𝐶) = ∅ ∨ 𝑖 = 𝑗))
4 df-or 842 . . . . . . 7 (((𝐵𝐶) = ∅ ∨ 𝑖 = 𝑗) ↔ (¬ (𝐵𝐶) = ∅ → 𝑖 = 𝑗))
5 neq0 4306 . . . . . . . . . 10 (¬ (𝐵𝐶) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵𝐶))
6 elin 4166 . . . . . . . . . . 11 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
76exbii 1839 . . . . . . . . . 10 (∃𝑥 𝑥 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐵𝑥𝐶))
85, 7bitri 276 . . . . . . . . 9 (¬ (𝐵𝐶) = ∅ ↔ ∃𝑥(𝑥𝐵𝑥𝐶))
98imbi1i 351 . . . . . . . 8 ((¬ (𝐵𝐶) = ∅ → 𝑖 = 𝑗) ↔ (∃𝑥(𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
10 19.23v 1934 . . . . . . . 8 (∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) ↔ (∃𝑥(𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
119, 10bitr4i 279 . . . . . . 7 ((¬ (𝐵𝐶) = ∅ → 𝑖 = 𝑗) ↔ ∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
123, 4, 113bitri 298 . . . . . 6 ((𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ∀𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
1312ralbii 3162 . . . . 5 (∀𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ∀𝑗𝐴𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
14 ralcom4 3232 . . . . 5 (∀𝑗𝐴𝑥((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
1513, 14bitri 276 . . . 4 (∀𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ∀𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
1615ralbii 3162 . . 3 (∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ∀𝑖𝐴𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
17 disjor.1 . . . . . 6 (𝑖 = 𝑗𝐵 = 𝐶)
1817eleq2d 2895 . . . . 5 (𝑖 = 𝑗 → (𝑥𝐵𝑥𝐶))
1918rmo4 3718 . . . 4 (∃*𝑖𝐴 𝑥𝐵 ↔ ∀𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
2019albii 1811 . . 3 (∀𝑥∃*𝑖𝐴 𝑥𝐵 ↔ ∀𝑥𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
212, 16, 203bitr4i 304 . 2 (∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ∀𝑥∃*𝑖𝐴 𝑥𝐵)
221, 21bitr4i 279 1 (Disj 𝑖𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  wal 1526   = wceq 1528  wex 1771  wcel 2105  wral 3135  ∃*wrmo 3138  cin 3932  c0 4288  Disj wdisj 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rmo 3143  df-v 3494  df-dif 3936  df-in 3940  df-nul 4289  df-disj 5023
This theorem is referenced by:  disjors  5038  disjord  5045  disjiunb  5046  disjxiun  5054  disjxun  5055  otsndisj  5400  qsdisj2  8364  s3sndisj  14315  cshwsdisj  16420  dyadmbl  24128  numedglnl  26856  clwwlknondisj  27817  2wspmdisj  28043  disjnf  30248  disjorsf  30258  poimirlem26  34799  mblfinlem2  34811  ndisj2  41190  nnfoctbdjlem  42614  iundjiun  42619  otiunsndisjX  43355
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