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Mirrors > Home > MPE Home > Th. List > eq0ALT | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
eq0ALT | ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2729 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) | |
2 | noel 4289 | . . . 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) |
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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