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Theorem disjss2 3739
Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss2 (x A B𝐶 → (Disj x A 𝐶Disj x A B))

Proof of Theorem disjss2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ssel 2933 . . . . 5 (B𝐶 → (y By 𝐶))
21ralimi 2378 . . . 4 (x A B𝐶x A (y By 𝐶))
3 rmoim 2734 . . . 4 (x A (y By 𝐶) → (∃*x A y 𝐶∃*x A y B))
42, 3syl 14 . . 3 (x A B𝐶 → (∃*x A y 𝐶∃*x A y B))
54alimdv 1756 . 2 (x A B𝐶 → (y∃*x A y 𝐶y∃*x A y B))
6 df-disj 3737 . 2 (Disj x A 𝐶y∃*x A y 𝐶)
7 df-disj 3737 . 2 (Disj x A By∃*x A y B)
85, 6, 73imtr4g 194 1 (x A B𝐶 → (Disj x A 𝐶Disj x A B))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240   wcel 1390  wral 2300  ∃*wrmo 2303  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-bndl 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:  disjeq2  3740  0disj  3752
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