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Theorem disjeq1 5116
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 4042 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 disjss1 5115 . . 3 (𝐵𝐴 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
4 eqimss 4041 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 disjss1 5115 . . 3 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
73, 6impbid 212 1 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wss 3950  Disj wdisj 5109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-mo 2539  df-cleq 2728  df-clel 2815  df-rmo 3379  df-ss 3967  df-disj 5110
This theorem is referenced by:  disjeq1d  5117  volfiniun  25583  disjrnmpt  32599  iundisj2cnt  32802  unelldsys  34160  sigapildsys  34164  ldgenpisyslem1  34165  rossros  34182  measvun  34211  pmeasmono  34327  pmeasadd  34328  meadjuni  46477
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