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Theorem elfvne0 45752
Description: If a function value has a member, then the function is not an empty set (An artifact of our function value definition.) (Contributed by Zhi Wang, 16-Sep-2024.)
Assertion
Ref Expression
elfvne0 (𝐴 ∈ (𝐹𝐵) → 𝐹 ≠ ∅)

Proof of Theorem elfvne0
StepHypRef Expression
1 ne0i 4233 . 2 (𝐴 ∈ (𝐹𝐵) → (𝐹𝐵) ≠ ∅)
2 fveq1 6685 . . . 4 (𝐹 = ∅ → (𝐹𝐵) = (∅‘𝐵))
3 0fv 6725 . . . 4 (∅‘𝐵) = ∅
42, 3eqtrdi 2790 . . 3 (𝐹 = ∅ → (𝐹𝐵) = ∅)
54necon3i 2967 . 2 ((𝐹𝐵) ≠ ∅ → 𝐹 ≠ ∅)
61, 5syl 17 1 (𝐴 ∈ (𝐹𝐵) → 𝐹 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  wne 2935  c0 4221  cfv 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-ne 2936  df-ral 3059  df-rex 3060  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-dm 5545  df-iota 6307  df-fv 6357
This theorem is referenced by:  neircl  45767
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