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Theorem eq0ALT 4357
Description: Alternate proof of eq0 4356. Shorter, but requiring df-clel 2814, ax-8 2108. (Contributed by NM, 29-Aug-1993.) Avoid ax-11 2155, ax-12 2175. (Revised by GG 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 2728 . 2 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
2 noel 4344 . . . 4 ¬ 𝑥 ∈ ∅
32nbn 372 . . 3 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
43albii 1816 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
51, 4bitr4i 278 1 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wal 1535   = wceq 1537  wcel 2106  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-dif 3966  df-nul 4340
This theorem is referenced by: (None)
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