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Theorem disjorsf 28609
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 2750 . . 3 𝑖𝐵
3 nfcsb1v 3514 . . 3 𝑥𝑖 / 𝑥𝐵
4 csbeq1a 3507 . . 3 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
51, 2, 3, 4cbvdisjf 28601 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑖𝐴 𝑖 / 𝑥𝐵)
6 csbeq1 3501 . . 3 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
76disjor 4561 . 2 (Disj 𝑖𝐴 𝑖 / 𝑥𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
85, 7bitri 262 1 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wo 381   = wceq 1474  wnfc 2737  wral 2895  csb 3498  cin 3538  c0 3873  Disj wdisj 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rmo 2903  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-in 3546  df-nul 3874  df-disj 4548
This theorem is referenced by:  disjif2  28610  disjdsct  28697
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