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Theorem disjss1 3912
Description: A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss1 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem disjss1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssel 3091 . . . . . 6 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 334 . . . . 5 (𝐴𝐵 → ((𝑥𝐴𝑦𝐶) → (𝑥𝐵𝑦𝐶)))
32alrimiv 1846 . . . 4 (𝐴𝐵 → ∀𝑥((𝑥𝐴𝑦𝐶) → (𝑥𝐵𝑦𝐶)))
4 moim 2063 . . . 4 (∀𝑥((𝑥𝐴𝑦𝐶) → (𝑥𝐵𝑦𝐶)) → (∃*𝑥(𝑥𝐵𝑦𝐶) → ∃*𝑥(𝑥𝐴𝑦𝐶)))
53, 4syl 14 . . 3 (𝐴𝐵 → (∃*𝑥(𝑥𝐵𝑦𝐶) → ∃*𝑥(𝑥𝐴𝑦𝐶)))
65alimdv 1851 . 2 (𝐴𝐵 → (∀𝑦∃*𝑥(𝑥𝐵𝑦𝐶) → ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐶)))
7 dfdisj2 3908 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥(𝑥𝐵𝑦𝐶))
8 dfdisj2 3908 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐶))
96, 7, 83imtr4g 204 1 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1329  wcel 1480  ∃*wmo 2000  wss 3071  Disj wdisj 3906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-rmo 2424  df-in 3077  df-ss 3084  df-disj 3907
This theorem is referenced by:  disjeq1  3913  disjx0  3928  fsumiun  11258
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