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Theorem disjorsf 29621
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 2866 . . 3 𝑖𝐵
3 nfcsb1v 3655 . . 3 𝑥𝑖 / 𝑥𝐵
4 csbeq1a 3648 . . 3 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
51, 2, 3, 4cbvdisjf 29613 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑖𝐴 𝑖 / 𝑥𝐵)
6 csbeq1 3642 . . 3 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
76disjor 4742 . 2 (Disj 𝑖𝐴 𝑖 / 𝑥𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
85, 7bitri 264 1 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382   = wceq 1596  wnfc 2853  wral 3014  csb 3639  cin 3679  c0 4023  Disj wdisj 4728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rmo 3022  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-in 3687  df-nul 4024  df-disj 4729
This theorem is referenced by:  disjif2  29622  disjdsct  29710
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