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Theorem disjeq2 3740
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 2992 . . . 4 (B = 𝐶𝐶B)
21ralimi 2378 . . 3 (x A B = 𝐶x A 𝐶B)
3 disjss2 3739 . . 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 2991 . . . 4 (B = 𝐶B𝐶)
65ralimi 2378 . . 3 (x A B = 𝐶x A B𝐶)
7 disjss2 3739 . . 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 1242  wral 2300  wss 2911  Disj wdisj 3736
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-rmo 2308  df-in 2918  df-ss 2925  df-disj 3737
This theorem is referenced by:  disjeq2dv  3741
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