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Theorem bj-rest0 37076
Description: An elementwise intersection on a family containing the empty set contains the empty set. (Contributed by BJ, 27-Apr-2021.)
Assertion
Ref Expression
bj-rest0 ((𝑋𝑉𝐴𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋t 𝐴)))

Proof of Theorem bj-rest0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 in0 4401 . . . . 5 (𝐴 ∩ ∅) = ∅
2 incom 4217 . . . . 5 (𝐴 ∩ ∅) = (∅ ∩ 𝐴)
31, 2eqtr3i 2765 . . . 4 ∅ = (∅ ∩ 𝐴)
4 0ex 5313 . . . . 5 ∅ ∈ V
5 eleq1 2827 . . . . . 6 (𝑥 = ∅ → (𝑥𝑋 ↔ ∅ ∈ 𝑋))
6 ineq1 4221 . . . . . . 7 (𝑥 = ∅ → (𝑥𝐴) = (∅ ∩ 𝐴))
76eqeq2d 2746 . . . . . 6 (𝑥 = ∅ → (∅ = (𝑥𝐴) ↔ ∅ = (∅ ∩ 𝐴)))
85, 7anbi12d 632 . . . . 5 (𝑥 = ∅ → ((𝑥𝑋 ∧ ∅ = (𝑥𝐴)) ↔ (∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴))))
94, 8spcev 3606 . . . 4 ((∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴)) → ∃𝑥(𝑥𝑋 ∧ ∅ = (𝑥𝐴)))
103, 9mpan2 691 . . 3 (∅ ∈ 𝑋 → ∃𝑥(𝑥𝑋 ∧ ∅ = (𝑥𝐴)))
11 df-rex 3069 . . 3 (∃𝑥𝑋 ∅ = (𝑥𝐴) ↔ ∃𝑥(𝑥𝑋 ∧ ∅ = (𝑥𝐴)))
1210, 11sylibr 234 . 2 (∅ ∈ 𝑋 → ∃𝑥𝑋 ∅ = (𝑥𝐴))
13 elrest 17474 . 2 ((𝑋𝑉𝐴𝑊) → (∅ ∈ (𝑋t 𝐴) ↔ ∃𝑥𝑋 ∅ = (𝑥𝐴)))
1412, 13imbitrrid 246 1 ((𝑋𝑉𝐴𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋t 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wrex 3068  cin 3962  c0 4339  (class class class)co 7431  t crest 17467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17469
This theorem is referenced by: (None)
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