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Theorem eq0ALT 4303
Description: Alternate proof of eq0 4302. Shorter, but requiring df-clel 2814, ax-8 2108. (Contributed by NM, 29-Aug-1993.) Avoid ax-11 2154, ax-12 2171. (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 2729 . 2 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
2 noel 4289 . . . 4 ¬ 𝑥 ∈ ∅
32nbn 372 . . 3 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
43albii 1821 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
51, 4bitr4i 277 1 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1539   = wceq 1541  wcel 2106  c0 4281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-dif 3912  df-nul 4282
This theorem is referenced by: (None)
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