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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 4290 (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 2988). (Contributed by BJ, 19-Mar-2021.) |
Ref | Expression |
---|---|
ab0 | ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfab1 2957 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
2 | 1 | eq0f 4255 | . 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝑥 ∣ 𝜑}) |
3 | abid 2780 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
4 | 3 | notbii 323 | . . 3 ⊢ (¬ 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ¬ 𝜑) |
5 | 4 | albii 1821 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥 ¬ 𝜑) |
6 | 2, 5 | bitri 278 | 1 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∀wal 1536 = wceq 1538 ∈ wcel 2111 {cab 2776 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-dif 3884 df-nul 4244 |
This theorem is referenced by: dfnf5 4288 rab0 4291 rabeq0 4292 abfOLD 4311 0mpo0 7216 |
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