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Theorem bj-rest0 33273
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 4076 . . . . 5 (𝐴 ∩ ∅) = ∅
2 incom 3913 . . . . 5 (𝐴 ∩ ∅) = (∅ ∩ 𝐴)
31, 2eqtr3i 2748 . . . 4 ∅ = (∅ ∩ 𝐴)
4 0ex 4898 . . . . 5 ∅ ∈ V
5 eleq1 2791 . . . . . 6 (𝑥 = ∅ → (𝑥𝑋 ↔ ∅ ∈ 𝑋))
6 ineq1 3915 . . . . . . 7 (𝑥 = ∅ → (𝑥𝐴) = (∅ ∩ 𝐴))
76eqeq2d 2734 . . . . . 6 (𝑥 = ∅ → (∅ = (𝑥𝐴) ↔ ∅ = (∅ ∩ 𝐴)))
85, 7anbi12d 749 . . . . 5 (𝑥 = ∅ → ((𝑥𝑋 ∧ ∅ = (𝑥𝐴)) ↔ (∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴))))
94, 8spcev 3404 . . . 4 ((∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴)) → ∃𝑥(𝑥𝑋 ∧ ∅ = (𝑥𝐴)))
103, 9mpan2 709 . . 3 (∅ ∈ 𝑋 → ∃𝑥(𝑥𝑋 ∧ ∅ = (𝑥𝐴)))
11 df-rex 3020 . . 3 (∃𝑥𝑋 ∅ = (𝑥𝐴) ↔ ∃𝑥(𝑥𝑋 ∧ ∅ = (𝑥𝐴)))
1210, 11sylibr 224 . 2 (∅ ∈ 𝑋 → ∃𝑥𝑋 ∅ = (𝑥𝐴))
13 elrest 16211 . 2 ((𝑋𝑉𝐴𝑊) → (∅ ∈ (𝑋t 𝐴) ↔ ∃𝑥𝑋 ∅ = (𝑥𝐴)))
1412, 13syl5ibr 236 1 ((𝑋𝑉𝐴𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋t 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wex 1817  wcel 2103  wrex 3015  cin 3679  c0 4023  (class class class)co 6765  t crest 16204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-rest 16206
This theorem is referenced by: (None)
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