Theorem eq0ALT 4374

Description: Alternate proof of eq0 4373. Shorter, but requiring df-clel 2819, ax-8 2110. (Contributed by NM, 29-Aug-1993.) Avoid ax-11 2158, ax-12 2178. (Revised by GG 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 2733	. 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
2		noel 4360	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
3	2	nbn 372	. . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
4	3	albii 1817	. 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
5	1, 4	bitr4i 278	1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1535   = wceq 1537   ∈ wcel 2108  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-dif 3979  df-nul 4353

This theorem is referenced by: (None)