Theorem eq0ALT 4344

Description: Alternate proof of eq0 4343. Shorter, but requiring df-clel 2810, 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 ¬ 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem eq0ALT

Step	Hyp	Ref	Expression
1		dfcleq 2725	. 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
2		noel 4330	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
3	2	nbn 372	. . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
4	3	albii 1821	. 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
5	1, 4	bitr4i 277	1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∅c0 4322

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 2703

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 2710  df-cleq 2724  df-clel 2810  df-dif 3951  df-nul 4323

This theorem is referenced by: (None)