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Theorem disjors 5128
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 2901 . . 3 𝑖𝐵
2 nfcsb1v 3917 . . 3 𝑥𝑖 / 𝑥𝐵
3 csbeq1a 3906 . . 3 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
41, 2, 3cbvdisj 5122 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑖𝐴 𝑖 / 𝑥𝐵)
5 csbeq1 3895 . . 3 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
65disjor 5127 . 2 (Disj 𝑖𝐴 𝑖 / 𝑥𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
74, 6bitri 274 1 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 843   = wceq 1539  wral 3059  csb 3892  cin 3946  c0 4321  Disj wdisj 5112
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rmo 3374  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-in 3954  df-nul 4322  df-disj 5113
This theorem is referenced by:  disji2  5129  disjprgw  5142  disjprg  5143  disjxiun  5144  disjxun  5145  iundisj2  25298  disji2f  32075  disjpreima  32082  disjxpin  32086  iundisj2f  32088  disjunsn  32092  iundisj2fi  32275  disjxp1  44057  disjinfi  44189
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