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Theorem disjeq2 3723
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjeq2 (x A B = 𝐶 → (Disj x A BDisj x A 𝐶))

Proof of Theorem disjeq2
StepHypRef Expression
1 eqimss2 2975 . . . 4 (B = 𝐶𝐶B)
21ralimi 2362 . . 3 (x A B = 𝐶x A 𝐶B)
3 disjss2 3722 . . 3 (x A 𝐶B → (Disj x A BDisj x A 𝐶))
42, 3syl 14 . 2 (x A B = 𝐶 → (Disj x A BDisj x A 𝐶))
5 eqimss 2974 . . . 4 (B = 𝐶B𝐶)
65ralimi 2362 . . 3 (x A B = 𝐶x A B𝐶)
7 disjss2 3722 . . 3 (x A B𝐶 → (Disj x A 𝐶Disj x A B))
86, 7syl 14 . 2 (x A B = 𝐶 → (Disj x A 𝐶Disj x A B))
94, 8impbid 120 1 (x A B = 𝐶 → (Disj x A BDisj x A 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228  wral 2284  wss 2894  Disj wdisj 3719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-ral 2289  df-rmo 2292  df-in 2901  df-ss 2908  df-disj 3720
This theorem is referenced by:  disjeq2dv  3724
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