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Mirrors > Home > MPE Home > Th. List > ab0 | Structured version Visualization version GIF version |
Description: The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 4266 (from which it could be proved using as many essential proof steps but one fewer syntactic step, at the cost of depending on df-ne 2935). (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2811, ax-8 2115. (Revised by Gino Giotto, 30-Aug-2024.) (Proof shortened by SN, 8-Sep-2024.) |
Ref | Expression |
---|---|
ab0 | ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abbi 2805 | . 2 ⊢ (∀𝑥(𝜑 ↔ ⊥) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}) | |
2 | nbfal 1557 | . . 3 ⊢ (¬ 𝜑 ↔ (𝜑 ↔ ⊥)) | |
3 | 2 | albii 1826 | . 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥(𝜑 ↔ ⊥)) |
4 | dfnul4 4211 | . . 3 ⊢ ∅ = {𝑥 ∣ ⊥} | |
5 | 4 | eqeq2i 2751 | . 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}) |
6 | 1, 3, 5 | 3bitr4ri 307 | 1 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∀wal 1540 = wceq 1542 ⊥wfal 1554 {cab 2716 ∅c0 4209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-dif 3844 df-nul 4210 |
This theorem is referenced by: dfnf5 4263 abn0 4266 rab0 4268 rabeq0 4270 abfOLD 4289 0mpo0 7245 fsetdmprc0 8458 |
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