MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjeq1 Structured version   Visualization version   GIF version

Theorem disjeq1 4904
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjeq1 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem disjeq1
StepHypRef Expression
1 eqimss2 3914 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 disjss1 4903 . . 3 (𝐵𝐴 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
4 eqimss 3913 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 disjss1 4903 . . 3 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
73, 6impbid 204 1 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1507  wss 3829  Disj wdisj 4897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-clab 2759  df-cleq 2771  df-clel 2846  df-rmo 3096  df-in 3836  df-ss 3843  df-disj 4898
This theorem is referenced by:  disjeq1d  4905  volfiniun  23851  disjrnmpt  30101  iundisj2cnt  30278  unelldsys  31068  sigapildsys  31072  ldgenpisyslem1  31073  rossros  31090  measvun  31119  pmeasmono  31233  pmeasadd  31234  meadjuni  42176
  Copyright terms: Public domain W3C validator