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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 4314 (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 2944). (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2816, ax-8 2108. (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 2810 | . 2 ⊢ (∀𝑥(𝜑 ↔ ⊥) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}) | |
2 | nbfal 1554 | . . 3 ⊢ (¬ 𝜑 ↔ (𝜑 ↔ ⊥)) | |
3 | 2 | albii 1822 | . 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥(𝜑 ↔ ⊥)) |
4 | dfnul4 4258 | . . 3 ⊢ ∅ = {𝑥 ∣ ⊥} | |
5 | 4 | eqeq2i 2751 | . 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}) |
6 | 1, 3, 5 | 3bitr4ri 304 | 1 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1537 = wceq 1539 ⊥wfal 1551 {cab 2715 ∅c0 4256 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-dif 3890 df-nul 4257 |
This theorem is referenced by: dfnf5 4311 abn0 4314 rab0 4316 rabeq0 4318 abfOLD 4337 sticksstones22 40124 |
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