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Theorem disjorsf 32593
Description: Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
Hypothesis
Ref Expression
disjorsf.1 𝑥𝐴
Assertion
Ref Expression
disjorsf (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
Distinct variable groups:   𝑖,𝑗,𝑥   𝐴,𝑖,𝑗   𝐵,𝑖,𝑗
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem disjorsf
StepHypRef Expression
1 disjorsf.1 . . 3 𝑥𝐴
2 nfcv 2905 . . 3 𝑖𝐵
3 nfcsb1v 3923 . . 3 𝑥𝑖 / 𝑥𝐵
4 csbeq1a 3913 . . 3 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
51, 2, 3, 4cbvdisjf 32584 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑖𝐴 𝑖 / 𝑥𝐵)
6 csbeq1 3902 . . 3 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
76disjor 5125 . 2 (Disj 𝑖𝐴 𝑖 / 𝑥𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
85, 7bitri 275 1 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wo 848   = wceq 1540  wnfc 2890  wral 3061  csb 3899  cin 3950  c0 4333  Disj wdisj 5110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rmo 3380  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-in 3958  df-nul 4334  df-disj 5111
This theorem is referenced by:  disjif2  32594  disjdsct  32712
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