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Theorem elfvdmOLD 6529
 Description: Obsolete proof of elfvdm 6528 as of 22-Oct-2022. (Contributed by NM, 12-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elfvdmOLD (𝐴 ∈ (𝐹𝐵) → 𝐵 ∈ dom 𝐹)

Proof of Theorem elfvdmOLD
StepHypRef Expression
1 ne0i 4180 . 2 (𝐴 ∈ (𝐹𝐵) → (𝐹𝐵) ≠ ∅)
2 ndmfv 6526 . . 3 𝐵 ∈ dom 𝐹 → (𝐹𝐵) = ∅)
32necon1ai 2987 . 2 ((𝐹𝐵) ≠ ∅ → 𝐵 ∈ dom 𝐹)
41, 3syl 17 1 (𝐴 ∈ (𝐹𝐵) → 𝐵 ∈ dom 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2051   ≠ wne 2960  ∅c0 4172  dom cdm 5403  ‘cfv 6185 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743  ax-nul 5063  ax-pow 5115 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-dm 5413  df-iota 6149  df-fv 6193 This theorem is referenced by: (None)
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