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Theorem eq0ALT 4345
Description: Alternate proof of eq0 4344. Shorter, but requiring df-clel 2811, ax-8 2109. (Contributed by NM, 29-Aug-1993.) Avoid ax-11 2155, ax-12 2172. (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
eq0ALT (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem eq0ALT
StepHypRef Expression
1 dfcleq 2726 . 2 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
2 noel 4331 . . . 4 ¬ 𝑥 ∈ ∅
32nbn 373 . . 3 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
43albii 1822 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
51, 4bitr4i 278 1 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1540   = wceq 1542  wcel 2107  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-dif 3952  df-nul 4324
This theorem is referenced by: (None)
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