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Theorem bj-rest0 34278
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 4342 . . . . 5 (𝐴 ∩ ∅) = ∅
2 incom 4175 . . . . 5 (𝐴 ∩ ∅) = (∅ ∩ 𝐴)
31, 2eqtr3i 2843 . . . 4 ∅ = (∅ ∩ 𝐴)
4 0ex 5202 . . . . 5 ∅ ∈ V
5 eleq1 2897 . . . . . 6 (𝑥 = ∅ → (𝑥𝑋 ↔ ∅ ∈ 𝑋))
6 ineq1 4178 . . . . . . 7 (𝑥 = ∅ → (𝑥𝐴) = (∅ ∩ 𝐴))
76eqeq2d 2829 . . . . . 6 (𝑥 = ∅ → (∅ = (𝑥𝐴) ↔ ∅ = (∅ ∩ 𝐴)))
85, 7anbi12d 630 . . . . 5 (𝑥 = ∅ → ((𝑥𝑋 ∧ ∅ = (𝑥𝐴)) ↔ (∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴))))
94, 8spcev 3604 . . . 4 ((∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴)) → ∃𝑥(𝑥𝑋 ∧ ∅ = (𝑥𝐴)))
103, 9mpan2 687 . . 3 (∅ ∈ 𝑋 → ∃𝑥(𝑥𝑋 ∧ ∅ = (𝑥𝐴)))
11 df-rex 3141 . . 3 (∃𝑥𝑋 ∅ = (𝑥𝐴) ↔ ∃𝑥(𝑥𝑋 ∧ ∅ = (𝑥𝐴)))
1210, 11sylibr 235 . 2 (∅ ∈ 𝑋 → ∃𝑥𝑋 ∅ = (𝑥𝐴))
13 elrest 16689 . 2 ((𝑋𝑉𝐴𝑊) → (∅ ∈ (𝑋t 𝐴) ↔ ∃𝑥𝑋 ∅ = (𝑥𝐴)))
1412, 13syl5ibr 247 1 ((𝑋𝑉𝐴𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋t 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  wrex 3136  cin 3932  c0 4288  (class class class)co 7145  t crest 16682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-rest 16684
This theorem is referenced by: (None)
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