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Theorem disjors 5014
 Description: Two ways to say that a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 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 2920 . . 3 𝑖𝐵
2 nfcsb1v 3830 . . 3 𝑥𝑖 / 𝑥𝐵
3 csbeq1a 3820 . . 3 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
41, 2, 3cbvdisj 5008 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑖𝐴 𝑖 / 𝑥𝐵)
5 csbeq1 3809 . . 3 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
65disjor 5013 . 2 (Disj 𝑖𝐴 𝑖 / 𝑥𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
74, 6bitri 278 1 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∨ wo 845   = wceq 1539  ∀wral 3071  ⦋csb 3806   ∩ cin 3858  ∅c0 4226  Disj wdisj 4998 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rmo 3079  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-in 3866  df-nul 4227  df-disj 4999 This theorem is referenced by:  disji2  5015  disjprgw  5028  disjprg  5029  disjxiun  5030  disjxun  5031  iundisj2  24250  disji2f  30439  disjpreima  30446  disjxpin  30450  iundisj2f  30452  disjunsn  30456  iundisj2fi  30643  disjxp1  42077  disjinfi  42191
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