Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ralf0 | Structured version Visualization version GIF version |
Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) |
Ref | Expression |
---|---|
ralf0.1 | ⊢ ¬ 𝜑 |
Ref | Expression |
---|---|
ralf0 | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralf0.1 | . . . 4 ⊢ ¬ 𝜑 | |
2 | mtt 354 | . . . 4 ⊢ (¬ 𝜑 → (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 → 𝜑))) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 → 𝜑)) |
4 | 3 | albii 1747 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
5 | eq0 3929 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
6 | df-ral 2917 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
7 | 4, 5, 6 | 3bitr4ri 293 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∀wal 1481 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∅c0 3915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-v 3202 df-dif 3577 df-nul 3916 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |