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Mirrors > Home > MPE Home > Th. List > Mathboxes > ralfal | Structured version Visualization version GIF version |
Description: Two ways of expressing empty set. (Contributed by Glauco Siliprandi, 24-Jan-2024.) |
Ref | Expression |
---|---|
ralfal.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
ralfal | ⊢ (𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 ⊥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fal 1546 | . . . 4 ⊢ (⊥ ↔ ¬ ⊤) | |
2 | 1 | ralbii 3085 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ⊥ ↔ ∀𝑥 ∈ 𝐴 ¬ ⊤) |
3 | ralnex 3064 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ ⊤ ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤) | |
4 | 2, 3 | bitri 275 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ⊥ ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤) |
5 | rextru 3069 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤) | |
6 | 5 | notbii 320 | . 2 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤) |
7 | ralfal.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
8 | 7 | neq0f 4334 | . . 3 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) |
9 | 8 | con1bii 356 | . 2 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅) |
10 | 4, 6, 9 | 3bitr2ri 300 | 1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 ⊥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1533 ⊤wtru 1534 ⊥wfal 1545 ∃wex 1773 ∈ wcel 2098 Ⅎwnfc 2875 ∀wral 3053 ∃wrex 3062 ∅c0 4315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-dif 3944 df-nul 4316 |
This theorem is referenced by: (None) |
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