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Theorem disjnims 3981
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 2312 . . 3 𝑖𝐵
2 nfcsb1v 3082 . . 3 𝑥𝑖 / 𝑥𝐵
3 csbeq1a 3058 . . 3 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
41, 2, 3cbvdisj 3976 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑖𝐴 𝑖 / 𝑥𝐵)
5 csbeq1 3052 . . 3 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
65disjnim 3980 . 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 1348  wne 2340  wral 2448  csb 3049  cin 3120  c0 3414  Disj wdisj 3966
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-in 3127  df-nul 3415  df-disj 3967
This theorem is referenced by:  disji2  3982
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