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Mirrors > Home > MPE Home > Th. List > ralf0OLD | Structured version Visualization version GIF version |
Description: Obsolete proof of ralf0 4078 as of 14-Jul-2021. (Contributed by NM, 26-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
ralf0.1 | ⊢ ¬ 𝜑 |
Ref | Expression |
---|---|
ralf0OLD | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralf0.1 | . . . . 5 ⊢ ¬ 𝜑 | |
2 | con3 149 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (¬ 𝜑 → ¬ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | mpi 20 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → ¬ 𝑥 ∈ 𝐴) |
4 | 3 | alimi 1739 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
5 | df-ral 2917 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
6 | eq0 3929 | . . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
7 | 4, 5, 6 | 3imtr4i 281 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = ∅) |
8 | rzal 4073 | . 2 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) | |
9 | 7, 8 | impbii 199 | 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-ne 2795 df-ral 2917 df-v 3202 df-dif 3577 df-nul 3916 |
This theorem is referenced by: (None) |
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