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Theorem disjnims 3959
Description: If a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Jim Kingdon, 7-Oct-2022.)
Assertion
Ref Expression
disjnims (Disj 𝑥𝐴 𝐵 → ∀𝑖𝐴𝑗𝐴 (𝑖𝑗 → (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
Distinct variable groups:   𝑖,𝑗,𝑥,𝐴   𝐵,𝑖,𝑗
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disjnims
StepHypRef Expression
1 nfcv 2299 . . 3 𝑖𝐵
2 nfcsb1v 3064 . . 3 𝑥𝑖 / 𝑥𝐵
3 csbeq1a 3040 . . 3 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
41, 2, 3cbvdisj 3954 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑖𝐴 𝑖 / 𝑥𝐵)
5 csbeq1 3034 . . 3 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
65disjnim 3958 . 2 (Disj 𝑖𝐴 𝑖 / 𝑥𝐵 → ∀𝑖𝐴𝑗𝐴 (𝑖𝑗 → (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
74, 6sylbi 120 1 (Disj 𝑥𝐴 𝐵 → ∀𝑖𝐴𝑗𝐴 (𝑖𝑗 → (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  wne 2327  wral 2435  csb 3031  cin 3101  c0 3395  Disj wdisj 3944
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-in 3108  df-nul 3396  df-disj 3945
This theorem is referenced by:  disji2  3960
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