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Mirrors > Home > MPE Home > Th. List > ralf0OLD | Structured version Visualization version GIF version |
Description: Obsolete version of ralf0 4458 as of 2-Sep-2024. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ralf0OLD.1 | ⊢ ¬ 𝜑 |
Ref | Expression |
---|---|
ralf0OLD | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralf0OLD.1 | . . . 4 ⊢ ¬ 𝜑 | |
2 | mtt 364 | . . . 4 ⊢ (¬ 𝜑 → (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 → 𝜑))) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 → 𝜑)) |
4 | 3 | albii 1820 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
5 | eq0 4290 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
6 | df-ral 3062 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
7 | 4, 5, 6 | 3bitr4ri 303 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1538 = wceq 1540 ∈ 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: (None) |
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