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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 1553 | . . . 4 ⊢ (⊥ ↔ ¬ ⊤) | |
| 2 | 1 | ralbii 3093 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ⊥ ↔ ∀𝑥 ∈ 𝐴 ¬ ⊤) | 
| 3 | ralnex 3072 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ ⊤ ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤) | |
| 4 | 2, 3 | bitri 275 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ⊥ ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤) | 
| 5 | rextru 3077 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤) | |
| 6 | 5 | notbii 320 | . 2 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤) | 
| 7 | ralfal.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 8 | 7 | neq0f 4348 | . . 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 206 = wceq 1540 ⊤wtru 1541 ⊥wfal 1552 ∃wex 1779 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 ∃wrex 3070 ∅c0 4333 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-dif 3954 df-nul 4334 | 
| This theorem is referenced by: (None) | 
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