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Theorem bj-rest0 34503
 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 4302 . . . . 5 (𝐴 ∩ ∅) = ∅
2 incom 4131 . . . . 5 (𝐴 ∩ ∅) = (∅ ∩ 𝐴)
31, 2eqtr3i 2826 . . . 4 ∅ = (∅ ∩ 𝐴)
4 0ex 5178 . . . . 5 ∅ ∈ V
5 eleq1 2880 . . . . . 6 (𝑥 = ∅ → (𝑥𝑋 ↔ ∅ ∈ 𝑋))
6 ineq1 4134 . . . . . . 7 (𝑥 = ∅ → (𝑥𝐴) = (∅ ∩ 𝐴))
76eqeq2d 2812 . . . . . 6 (𝑥 = ∅ → (∅ = (𝑥𝐴) ↔ ∅ = (∅ ∩ 𝐴)))
85, 7anbi12d 633 . . . . 5 (𝑥 = ∅ → ((𝑥𝑋 ∧ ∅ = (𝑥𝐴)) ↔ (∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴))))
94, 8spcev 3558 . . . 4 ((∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴)) → ∃𝑥(𝑥𝑋 ∧ ∅ = (𝑥𝐴)))
103, 9mpan2 690 . . 3 (∅ ∈ 𝑋 → ∃𝑥(𝑥𝑋 ∧ ∅ = (𝑥𝐴)))
11 df-rex 3115 . . 3 (∃𝑥𝑋 ∅ = (𝑥𝐴) ↔ ∃𝑥(𝑥𝑋 ∧ ∅ = (𝑥𝐴)))
1210, 11sylibr 237 . 2 (∅ ∈ 𝑋 → ∃𝑥𝑋 ∅ = (𝑥𝐴))
13 elrest 16696 . 2 ((𝑋𝑉𝐴𝑊) → (∅ ∈ (𝑋t 𝐴) ↔ ∃𝑥𝑋 ∅ = (𝑥𝐴)))
1412, 13syl5ibr 249 1 ((𝑋𝑉𝐴𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋t 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2112  ∃wrex 3110   ∩ cin 3883  ∅c0 4246  (class class class)co 7139   ↾t crest 16689 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-rest 16691 This theorem is referenced by: (None)
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