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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 4376 (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 2936). (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2805, ax-8 2101. (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 | abbib 2799 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥} ↔ ∀𝑥(𝜑 ↔ ⊥)) | |
2 | dfnul4 4320 | . . 3 ⊢ ∅ = {𝑥 ∣ ⊥} | |
3 | 2 | eqeq2i 2740 | . 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}) |
4 | nbfal 1549 | . . 3 ⊢ (¬ 𝜑 ↔ (𝜑 ↔ ⊥)) | |
5 | 4 | albii 1814 | . 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥(𝜑 ↔ ⊥)) |
6 | 1, 3, 5 | 3bitr4i 303 | 1 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1532 = wceq 1534 ⊥wfal 1546 {cab 2704 ∅c0 4318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-dif 3947 df-nul 4319 |
This theorem is referenced by: dfnf5 4373 abn0 4376 rab0 4378 rabeq0 4380 abfOLD 4399 sticksstones22 41624 |
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