Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > falseral0 | Structured version Visualization version GIF version |
Description: A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.) |
Ref | Expression |
---|---|
falseral0 | ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3062 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
2 | 19.26 1872 | . . 3 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))) | |
3 | con3 153 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (¬ 𝜑 → ¬ 𝑥 ∈ 𝐴)) | |
4 | 3 | impcom 408 | . . . . . 6 ⊢ ((¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → ¬ 𝑥 ∈ 𝐴) |
5 | 4 | alimi 1812 | . . . . 5 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
6 | alnex 1782 | . . . . 5 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴) | |
7 | 5, 6 | sylib 217 | . . . 4 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → ¬ ∃𝑥 𝑥 ∈ 𝐴) |
8 | notnotb 314 | . . . . 5 ⊢ (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅) | |
9 | neq0 4292 | . . . . 5 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
10 | 8, 9 | xchbinx 333 | . . . 4 ⊢ (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴) |
11 | 7, 10 | sylibr 233 | . . 3 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → 𝐴 = ∅) |
12 | 2, 11 | sylbir 234 | . 2 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝐴 = ∅) |
13 | 1, 12 | sylan2b 594 | 1 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∀wal 1538 = wceq 1540 ∃wex 1780 ∈ wcel 2105 ∀wral 3061 ∅c0 4269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-ral 3062 df-dif 3901 df-nul 4270 |
This theorem is referenced by: uvtx01vtx 28053 |
Copyright terms: Public domain | W3C validator |