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Theorem disjors 5038
Description: Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjors (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
Distinct variable groups:   𝑖,𝑗,𝑥,𝐴   𝐵,𝑖,𝑗
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disjors
StepHypRef Expression
1 nfcv 2974 . . 3 𝑖𝐵
2 nfcsb1v 3904 . . 3 𝑥𝑖 / 𝑥𝐵
3 csbeq1a 3894 . . 3 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
41, 2, 3cbvdisj 5032 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑖𝐴 𝑖 / 𝑥𝐵)
5 csbeq1 3883 . . 3 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
65disjor 5037 . 2 (Disj 𝑖𝐴 𝑖 / 𝑥𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
74, 6bitri 276 1 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wo 841   = wceq 1528  wral 3135  csb 3880  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-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-in 3940  df-nul 4289  df-disj 5023
This theorem is referenced by:  disji2  5039  disjprgw  5052  disjprg  5053  disjxiun  5054  disjxun  5055  iundisj2  24077  disji2f  30255  disjpreima  30262  disjxpin  30266  iundisj2f  30268  disjunsn  30272  iundisj2fi  30446  disjxp1  41208  disjinfi  41330
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