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Theorem disjors 4667
Description: Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjors (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
Distinct variable groups:   𝑖,𝑗,𝑥,𝐴   𝐵,𝑖,𝑗
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disjors
StepHypRef Expression
1 nfcv 2793 . . 3 𝑖𝐵
2 nfcsb1v 3582 . . 3 𝑥𝑖 / 𝑥𝐵
3 csbeq1a 3575 . . 3 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
41, 2, 3cbvdisj 4662 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑖𝐴 𝑖 / 𝑥𝐵)
5 csbeq1 3569 . . 3 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
65disjor 4666 . 2 (Disj 𝑖𝐴 𝑖 / 𝑥𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
74, 6bitri 264 1 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382   = wceq 1523  wral 2941  csb 3566  cin 3606  c0 3948  Disj wdisj 4652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-in 3614  df-nul 3949  df-disj 4653
This theorem is referenced by:  disji2  4668  disjprg  4680  disjxiun  4681  disjxiunOLD  4682  disjxun  4683  iundisj2  23363  disji2f  29516  disjpreima  29523  disjxpin  29527  iundisj2f  29529  disjunsn  29533  iundisj2fi  29684  disjxp1  39552  disjinfi  39694
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