![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ab0 | Structured version Visualization version GIF version |
Description: The class of sets verifying a falsity is the empty set (closed form of abf 4236). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ab0 | ⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1869 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) | |
2 | stdpc4 2019 | . . . . 5 ⊢ (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑) | |
3 | sbn 2214 | . . . . 5 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) | |
4 | 2, 3 | sylib 210 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑) |
5 | df-clab 2753 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
6 | 4, 5 | sylnibr 321 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑}) |
7 | 1, 6 | alrimih 1786 | . 2 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑}) |
8 | eq0 4188 | . 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
9 | 7, 8 | sylibr 226 | 1 ⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1505 = wceq 1507 [wsb 2015 ∈ wcel 2050 {cab 2752 ∅c0 4172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-dif 3826 df-nul 4173 |
This theorem is referenced by: bj-abf 33717 bj-csbprc 33718 |
Copyright terms: Public domain | W3C validator |