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Theorem disjeq1 5078
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 4002 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 disjss1 5077 . . 3 (𝐵𝐴 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
4 eqimss 4001 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 disjss1 5077 . . 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 1542  wss 3911  Disj wdisj 5071
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-rmo 3352  df-v 3446  df-in 3918  df-ss 3928  df-disj 5072
This theorem is referenced by:  disjeq1d  5079  volfiniun  24927  disjrnmpt  31549  iundisj2cnt  31749  unelldsys  32814  sigapildsys  32818  ldgenpisyslem1  32819  rossros  32836  measvun  32865  pmeasmono  32981  pmeasadd  32982  meadjuni  44784
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