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Theorem disjor 4768
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 4755 . 2 (Disj 𝑖𝐴 𝐵 ↔ ∀𝑥∃*𝑖𝐴 𝑥𝐵)
2 ralcom4 3375 . . 3 (∀𝑖𝐴𝑥𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
3 orcom 849 . . . . . . 7 ((𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ((𝐵𝐶) = ∅ ∨ 𝑖 = 𝑗))
4 df-or 827 . . . . . . 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 3375 . . . . 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 disjor.1 . . . . . 6 (𝑖 = 𝑗𝐵 = 𝐶)
1817eleq2d 2836 . . . . 5 (𝑖 = 𝑗 → (𝑥𝐵𝑥𝐶))
1918rmo4 3551 . . . 4 (∃*𝑖𝐴 𝑥𝐵 ↔ ∀𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
2019albii 1895 . . 3 (∀𝑥∃*𝑖𝐴 𝑥𝐵 ↔ ∀𝑥𝑖𝐴𝑗𝐴 ((𝑥𝐵𝑥𝐶) → 𝑖 = 𝑗))
212, 16, 203bitr4i 292 . 2 (∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅) ↔ ∀𝑥∃*𝑖𝐴 𝑥𝐵)
221, 21bitr4i 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 826  wal 1629   = wceq 1631  wex 1852  wcel 2145  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 827  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:  disjors  4769  disjord  4775  disjiunb  4776  disjxiun  4783  disjxun  4784  otsndisj  5112  otiunsndisj  5113  qsdisj2  7976  s3sndisj  13915  s3iunsndisj  13916  cshwsdisj  16011  dyadmbl  23587  numedglnl  26260  clwwlknondisj  27286  clwwlknondisjOLD  27291  2wspmdisj  27518  disjnf  29719  disjorsf  29728  poimirlem26  33764  mblfinlem2  33776  ndisj2  39735  nnfoctbdjlem  41185  iundjiun  41190  otiunsndisjX  41817
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