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| Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by BJ, 15-Jul-2021.) | 
| Ref | Expression | 
|---|---|
| eq0f.1 | ⊢ Ⅎ𝑥𝐴 | 
| Ref | Expression | 
|---|---|
| eq0f | ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eq0f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 2 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑥∅ | |
| 3 | 1, 2 | cleqf 2934 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) | 
| 4 | noel 4338 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
| 5 | 4 | nbn 372 | . . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) | 
| 6 | 5 | albii 1819 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) | 
| 7 | 3, 6 | bitr4i 278 | 1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ∅c0 4333 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-dif 3954 df-nul 4334 | 
| This theorem is referenced by: neq0f 4348 ab0ALT 4381 bnj1476 34861 stoweidlem34 46049 | 
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