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Theorem disjeq1 5124
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 4041 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 disjss1 5123 . . 3 (𝐵𝐴 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
4 eqimss 4040 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 disjss1 5123 . . 3 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
73, 6impbid 211 1 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wss 3949  Disj wdisj 5117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-mo 2529  df-clab 2706  df-cleq 2720  df-clel 2806  df-rmo 3374  df-v 3475  df-in 3956  df-ss 3966  df-disj 5118
This theorem is referenced by:  disjeq1d  5125  volfiniun  25496  disjrnmpt  32396  iundisj2cnt  32588  unelldsys  33810  sigapildsys  33814  ldgenpisyslem1  33815  rossros  33832  measvun  33861  pmeasmono  33977  pmeasadd  33978  meadjuni  45874
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