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Theorem bj-rest0 37094
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 4395 . . . . 5 (𝐴 ∩ ∅) = ∅
2 incom 4209 . . . . 5 (𝐴 ∩ ∅) = (∅ ∩ 𝐴)
31, 2eqtr3i 2767 . . . 4 ∅ = (∅ ∩ 𝐴)
4 0ex 5307 . . . . 5 ∅ ∈ V
5 eleq1 2829 . . . . . 6 (𝑥 = ∅ → (𝑥𝑋 ↔ ∅ ∈ 𝑋))
6 ineq1 4213 . . . . . . 7 (𝑥 = ∅ → (𝑥𝐴) = (∅ ∩ 𝐴))
76eqeq2d 2748 . . . . . 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 3071 . . 3 (∃𝑥𝑋 ∅ = (𝑥𝐴) ↔ ∃𝑥(𝑥𝑋 ∧ ∅ = (𝑥𝐴)))
1210, 11sylibr 234 . 2 (∅ ∈ 𝑋 → ∃𝑥𝑋 ∅ = (𝑥𝐴))
13 elrest 17472 . 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 1540  wex 1779  wcel 2108  wrex 3070  cin 3950  c0 4333  (class class class)co 7431  t crest 17465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17467
This theorem is referenced by: (None)
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