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Mirrors > Home > MPE Home > Th. List > eq0f | Structured version Visualization version GIF version |
Description: The empty set 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 2751 | . . 3 ⊢ Ⅎ𝑥∅ | |
3 | 1, 2 | cleqf 2776 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
4 | noel 3878 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
5 | 4 | nbn 361 | . . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
6 | 5 | albii 1737 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
7 | 3, 6 | bitr4i 266 | 1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∀wal 1473 = wceq 1475 ∈ wcel 1977 Ⅎwnfc 2738 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-nul 3875 |
This theorem is referenced by: neq0f 3885 eq0 3888 ab0 3905 bnj1476 30171 stoweidlem34 38927 |
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