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Theorem disjorsf 30441
 Description: Two ways to say that a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 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 2919 . . 3 𝑖𝐵
3 nfcsb1v 3829 . . 3 𝑥𝑖 / 𝑥𝐵
4 csbeq1a 3819 . . 3 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
51, 2, 3, 4cbvdisjf 30432 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑖𝐴 𝑖 / 𝑥𝐵)
6 csbeq1 3808 . . 3 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
76disjor 5012 . 2 (Disj 𝑖𝐴 𝑖 / 𝑥𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
85, 7bitri 278 1 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∨ wo 844   = wceq 1538  Ⅎwnfc 2899  ∀wral 3070  ⦋csb 3805   ∩ cin 3857  ∅c0 4225  Disj wdisj 4997 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rmo 3078  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-in 3865  df-nul 4226  df-disj 4998 This theorem is referenced by:  disjif2  30442  disjdsct  30559
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