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Theorem disjors 4406
 Description: Two ways to say that a collection for is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjors Disj
Distinct variable groups:   ,,,   ,,
Allowed substitution hint:   ()

Proof of Theorem disjors
StepHypRef Expression
1 nfcv 2584 . . 3
2 nfcsb1v 3411 . . 3
3 csbeq1a 3404 . . 3
41, 2, 3cbvdisj 4401 . 2 Disj Disj
5 csbeq1 3398 . . 3
65disjor 4405 . 2 Disj
74, 6bitri 252 1 Disj
 Colors of variables: wff setvar class Syntax hints:   wb 187   wo 369   wceq 1437  wral 2775  csb 3395   cin 3435  c0 3761  Disj wdisj 4391 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-in 3443  df-nul 3762  df-disj 4392 This theorem is referenced by:  disji2  4407  disjprg  4416  disjxiun  4417  disjxun  4418  iundisj2  22488  disji2f  28176  disjpreima  28183  disjxpin  28187  iundisj2f  28189  disjunsn  28193  iundisj2fi  28366  disjxp1  37273  disjinfi  37317
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